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Theorem difinfsn 7098
Description: An infinite set minus one element is infinite. We require that the set has decidable equality. (Contributed by Jim Kingdon, 8-Aug-2023.)
Assertion
Ref Expression
difinfsn ((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) β†’ Ο‰ β‰Ό (𝐴 βˆ– {𝐡}))
Distinct variable groups:   π‘₯,𝐴,𝑦   π‘₯,𝐡,𝑦

Proof of Theorem difinfsn
Dummy variables π‘Ž 𝑓 𝑔 𝑛 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omp1eom 7093 . . . . 5 (Ο‰ βŠ” 1o) β‰ˆ Ο‰
2 simp2 998 . . . . 5 ((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) β†’ Ο‰ β‰Ό 𝐴)
3 endomtr 6789 . . . . 5 (((Ο‰ βŠ” 1o) β‰ˆ Ο‰ ∧ Ο‰ β‰Ό 𝐴) β†’ (Ο‰ βŠ” 1o) β‰Ό 𝐴)
41, 2, 3sylancr 414 . . . 4 ((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) β†’ (Ο‰ βŠ” 1o) β‰Ό 𝐴)
5 brdomi 6748 . . . 4 ((Ο‰ βŠ” 1o) β‰Ό 𝐴 β†’ βˆƒπ‘“ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴)
64, 5syl 14 . . 3 ((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) β†’ βˆƒπ‘“ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴)
7 inlresf1 7059 . . . . . . . 8 (inl β†Ύ Ο‰):ω–1-1β†’(Ο‰ βŠ” 1o)
8 f1co 5433 . . . . . . . 8 ((𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴 ∧ (inl β†Ύ Ο‰):ω–1-1β†’(Ο‰ βŠ” 1o)) β†’ (𝑓 ∘ (inl β†Ύ Ο‰)):ω–1-1→𝐴)
97, 8mpan2 425 . . . . . . 7 (𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴 β†’ (𝑓 ∘ (inl β†Ύ Ο‰)):ω–1-1→𝐴)
109ad2antlr 489 . . . . . 6 ((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) β†’ (𝑓 ∘ (inl β†Ύ Ο‰)):ω–1-1→𝐴)
11 f1f 5421 . . . . . . . . . . . 12 ((𝑓 ∘ (inl β†Ύ Ο‰)):ω–1-1→𝐴 β†’ (𝑓 ∘ (inl β†Ύ Ο‰)):Ο‰βŸΆπ΄)
1210, 11syl 14 . . . . . . . . . . 11 ((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) β†’ (𝑓 ∘ (inl β†Ύ Ο‰)):Ο‰βŸΆπ΄)
1312frnd 5375 . . . . . . . . . 10 ((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) β†’ ran (𝑓 ∘ (inl β†Ύ Ο‰)) βŠ† 𝐴)
1413sselda 3155 . . . . . . . . 9 (((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) ∧ 𝑠 ∈ ran (𝑓 ∘ (inl β†Ύ Ο‰))) β†’ 𝑠 ∈ 𝐴)
15 simpllr 534 . . . . . . . . . . . . . . . 16 ((((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) ∧ 𝑛 ∈ Ο‰) ∧ ((𝑓 ∘ (inl β†Ύ Ο‰))β€˜π‘›) = 𝐡) β†’ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡)
16 simpr 110 . . . . . . . . . . . . . . . 16 ((((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) ∧ 𝑛 ∈ Ο‰) ∧ ((𝑓 ∘ (inl β†Ύ Ο‰))β€˜π‘›) = 𝐡) β†’ ((𝑓 ∘ (inl β†Ύ Ο‰))β€˜π‘›) = 𝐡)
17 f1f 5421 . . . . . . . . . . . . . . . . . . . 20 ((inl β†Ύ Ο‰):ω–1-1β†’(Ο‰ βŠ” 1o) β†’ (inl β†Ύ Ο‰):Ο‰βŸΆ(Ο‰ βŠ” 1o))
187, 17ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (inl β†Ύ Ο‰):Ο‰βŸΆ(Ο‰ βŠ” 1o)
19 simpr 110 . . . . . . . . . . . . . . . . . . 19 (((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) ∧ 𝑛 ∈ Ο‰) β†’ 𝑛 ∈ Ο‰)
20 fvco3 5587 . . . . . . . . . . . . . . . . . . 19 (((inl β†Ύ Ο‰):Ο‰βŸΆ(Ο‰ βŠ” 1o) ∧ 𝑛 ∈ Ο‰) β†’ ((𝑓 ∘ (inl β†Ύ Ο‰))β€˜π‘›) = (π‘“β€˜((inl β†Ύ Ο‰)β€˜π‘›)))
2118, 19, 20sylancr 414 . . . . . . . . . . . . . . . . . 18 (((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) ∧ 𝑛 ∈ Ο‰) β†’ ((𝑓 ∘ (inl β†Ύ Ο‰))β€˜π‘›) = (π‘“β€˜((inl β†Ύ Ο‰)β€˜π‘›)))
2219fvresd 5540 . . . . . . . . . . . . . . . . . . 19 (((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) ∧ 𝑛 ∈ Ο‰) β†’ ((inl β†Ύ Ο‰)β€˜π‘›) = (inlβ€˜π‘›))
2322fveq2d 5519 . . . . . . . . . . . . . . . . . 18 (((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) ∧ 𝑛 ∈ Ο‰) β†’ (π‘“β€˜((inl β†Ύ Ο‰)β€˜π‘›)) = (π‘“β€˜(inlβ€˜π‘›)))
2421, 23eqtrd 2210 . . . . . . . . . . . . . . . . 17 (((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) ∧ 𝑛 ∈ Ο‰) β†’ ((𝑓 ∘ (inl β†Ύ Ο‰))β€˜π‘›) = (π‘“β€˜(inlβ€˜π‘›)))
2524adantr 276 . . . . . . . . . . . . . . . 16 ((((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) ∧ 𝑛 ∈ Ο‰) ∧ ((𝑓 ∘ (inl β†Ύ Ο‰))β€˜π‘›) = 𝐡) β†’ ((𝑓 ∘ (inl β†Ύ Ο‰))β€˜π‘›) = (π‘“β€˜(inlβ€˜π‘›)))
2615, 16, 253eqtr2rd 2217 . . . . . . . . . . . . . . 15 ((((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) ∧ 𝑛 ∈ Ο‰) ∧ ((𝑓 ∘ (inl β†Ύ Ο‰))β€˜π‘›) = 𝐡) β†’ (π‘“β€˜(inlβ€˜π‘›)) = (π‘“β€˜(inrβ€˜βˆ…)))
27 simp-4r 542 . . . . . . . . . . . . . . . 16 ((((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) ∧ 𝑛 ∈ Ο‰) ∧ ((𝑓 ∘ (inl β†Ύ Ο‰))β€˜π‘›) = 𝐡) β†’ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴)
28 djulcl 7049 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ Ο‰ β†’ (inlβ€˜π‘›) ∈ (Ο‰ βŠ” 1o))
2928ad2antlr 489 . . . . . . . . . . . . . . . 16 ((((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) ∧ 𝑛 ∈ Ο‰) ∧ ((𝑓 ∘ (inl β†Ύ Ο‰))β€˜π‘›) = 𝐡) β†’ (inlβ€˜π‘›) ∈ (Ο‰ βŠ” 1o))
30 0lt1o 6440 . . . . . . . . . . . . . . . . . 18 βˆ… ∈ 1o
31 djurcl 7050 . . . . . . . . . . . . . . . . . 18 (βˆ… ∈ 1o β†’ (inrβ€˜βˆ…) ∈ (Ο‰ βŠ” 1o))
3230, 31ax-mp 5 . . . . . . . . . . . . . . . . 17 (inrβ€˜βˆ…) ∈ (Ο‰ βŠ” 1o)
3332a1i 9 . . . . . . . . . . . . . . . 16 ((((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) ∧ 𝑛 ∈ Ο‰) ∧ ((𝑓 ∘ (inl β†Ύ Ο‰))β€˜π‘›) = 𝐡) β†’ (inrβ€˜βˆ…) ∈ (Ο‰ βŠ” 1o))
34 f1veqaeq 5769 . . . . . . . . . . . . . . . 16 ((𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴 ∧ ((inlβ€˜π‘›) ∈ (Ο‰ βŠ” 1o) ∧ (inrβ€˜βˆ…) ∈ (Ο‰ βŠ” 1o))) β†’ ((π‘“β€˜(inlβ€˜π‘›)) = (π‘“β€˜(inrβ€˜βˆ…)) β†’ (inlβ€˜π‘›) = (inrβ€˜βˆ…)))
3527, 29, 33, 34syl12anc 1236 . . . . . . . . . . . . . . 15 ((((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) ∧ 𝑛 ∈ Ο‰) ∧ ((𝑓 ∘ (inl β†Ύ Ο‰))β€˜π‘›) = 𝐡) β†’ ((π‘“β€˜(inlβ€˜π‘›)) = (π‘“β€˜(inrβ€˜βˆ…)) β†’ (inlβ€˜π‘›) = (inrβ€˜βˆ…)))
3626, 35mpd 13 . . . . . . . . . . . . . 14 ((((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) ∧ 𝑛 ∈ Ο‰) ∧ ((𝑓 ∘ (inl β†Ύ Ο‰))β€˜π‘›) = 𝐡) β†’ (inlβ€˜π‘›) = (inrβ€˜βˆ…))
3719adantr 276 . . . . . . . . . . . . . . . 16 ((((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) ∧ 𝑛 ∈ Ο‰) ∧ ((𝑓 ∘ (inl β†Ύ Ο‰))β€˜π‘›) = 𝐡) β†’ 𝑛 ∈ Ο‰)
38 djune 7076 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ Ο‰ ∧ βˆ… ∈ 1o) β†’ (inlβ€˜π‘›) β‰  (inrβ€˜βˆ…))
3937, 30, 38sylancl 413 . . . . . . . . . . . . . . 15 ((((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) ∧ 𝑛 ∈ Ο‰) ∧ ((𝑓 ∘ (inl β†Ύ Ο‰))β€˜π‘›) = 𝐡) β†’ (inlβ€˜π‘›) β‰  (inrβ€˜βˆ…))
4039neneqd 2368 . . . . . . . . . . . . . 14 ((((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) ∧ 𝑛 ∈ Ο‰) ∧ ((𝑓 ∘ (inl β†Ύ Ο‰))β€˜π‘›) = 𝐡) β†’ Β¬ (inlβ€˜π‘›) = (inrβ€˜βˆ…))
4136, 40pm2.65da 661 . . . . . . . . . . . . 13 (((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) ∧ 𝑛 ∈ Ο‰) β†’ Β¬ ((𝑓 ∘ (inl β†Ύ Ο‰))β€˜π‘›) = 𝐡)
4241ralrimiva 2550 . . . . . . . . . . . 12 ((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) β†’ βˆ€π‘› ∈ Ο‰ Β¬ ((𝑓 ∘ (inl β†Ύ Ο‰))β€˜π‘›) = 𝐡)
4312ffnd 5366 . . . . . . . . . . . . 13 ((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) β†’ (𝑓 ∘ (inl β†Ύ Ο‰)) Fn Ο‰)
44 eqeq1 2184 . . . . . . . . . . . . . . 15 (𝑠 = ((𝑓 ∘ (inl β†Ύ Ο‰))β€˜π‘›) β†’ (𝑠 = 𝐡 ↔ ((𝑓 ∘ (inl β†Ύ Ο‰))β€˜π‘›) = 𝐡))
4544notbid 667 . . . . . . . . . . . . . 14 (𝑠 = ((𝑓 ∘ (inl β†Ύ Ο‰))β€˜π‘›) β†’ (Β¬ 𝑠 = 𝐡 ↔ Β¬ ((𝑓 ∘ (inl β†Ύ Ο‰))β€˜π‘›) = 𝐡))
4645ralrn 5654 . . . . . . . . . . . . 13 ((𝑓 ∘ (inl β†Ύ Ο‰)) Fn Ο‰ β†’ (βˆ€π‘  ∈ ran (𝑓 ∘ (inl β†Ύ Ο‰)) Β¬ 𝑠 = 𝐡 ↔ βˆ€π‘› ∈ Ο‰ Β¬ ((𝑓 ∘ (inl β†Ύ Ο‰))β€˜π‘›) = 𝐡))
4743, 46syl 14 . . . . . . . . . . . 12 ((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) β†’ (βˆ€π‘  ∈ ran (𝑓 ∘ (inl β†Ύ Ο‰)) Β¬ 𝑠 = 𝐡 ↔ βˆ€π‘› ∈ Ο‰ Β¬ ((𝑓 ∘ (inl β†Ύ Ο‰))β€˜π‘›) = 𝐡))
4842, 47mpbird 167 . . . . . . . . . . 11 ((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) β†’ βˆ€π‘  ∈ ran (𝑓 ∘ (inl β†Ύ Ο‰)) Β¬ 𝑠 = 𝐡)
4948r19.21bi 2565 . . . . . . . . . 10 (((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) ∧ 𝑠 ∈ ran (𝑓 ∘ (inl β†Ύ Ο‰))) β†’ Β¬ 𝑠 = 𝐡)
50 velsn 3609 . . . . . . . . . 10 (𝑠 ∈ {𝐡} ↔ 𝑠 = 𝐡)
5149, 50sylnibr 677 . . . . . . . . 9 (((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) ∧ 𝑠 ∈ ran (𝑓 ∘ (inl β†Ύ Ο‰))) β†’ Β¬ 𝑠 ∈ {𝐡})
5214, 51eldifd 3139 . . . . . . . 8 (((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) ∧ 𝑠 ∈ ran (𝑓 ∘ (inl β†Ύ Ο‰))) β†’ 𝑠 ∈ (𝐴 βˆ– {𝐡}))
5352ex 115 . . . . . . 7 ((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) β†’ (𝑠 ∈ ran (𝑓 ∘ (inl β†Ύ Ο‰)) β†’ 𝑠 ∈ (𝐴 βˆ– {𝐡})))
5453ssrdv 3161 . . . . . 6 ((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) β†’ ran (𝑓 ∘ (inl β†Ύ Ο‰)) βŠ† (𝐴 βˆ– {𝐡}))
55 f1ssr 5428 . . . . . 6 (((𝑓 ∘ (inl β†Ύ Ο‰)):ω–1-1→𝐴 ∧ ran (𝑓 ∘ (inl β†Ύ Ο‰)) βŠ† (𝐴 βˆ– {𝐡})) β†’ (𝑓 ∘ (inl β†Ύ Ο‰)):ω–1-1β†’(𝐴 βˆ– {𝐡}))
5610, 54, 55syl2anc 411 . . . . 5 ((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) β†’ (𝑓 ∘ (inl β†Ύ Ο‰)):ω–1-1β†’(𝐴 βˆ– {𝐡}))
57 f1f 5421 . . . . . . 7 ((𝑓 ∘ (inl β†Ύ Ο‰)):ω–1-1β†’(𝐴 βˆ– {𝐡}) β†’ (𝑓 ∘ (inl β†Ύ Ο‰)):Ο‰βŸΆ(𝐴 βˆ– {𝐡}))
58 omex 4592 . . . . . . 7 Ο‰ ∈ V
59 fex 5745 . . . . . . 7 (((𝑓 ∘ (inl β†Ύ Ο‰)):Ο‰βŸΆ(𝐴 βˆ– {𝐡}) ∧ Ο‰ ∈ V) β†’ (𝑓 ∘ (inl β†Ύ Ο‰)) ∈ V)
6057, 58, 59sylancl 413 . . . . . 6 ((𝑓 ∘ (inl β†Ύ Ο‰)):ω–1-1β†’(𝐴 βˆ– {𝐡}) β†’ (𝑓 ∘ (inl β†Ύ Ο‰)) ∈ V)
61 f1eq1 5416 . . . . . . 7 (𝑔 = (𝑓 ∘ (inl β†Ύ Ο‰)) β†’ (𝑔:ω–1-1β†’(𝐴 βˆ– {𝐡}) ↔ (𝑓 ∘ (inl β†Ύ Ο‰)):ω–1-1β†’(𝐴 βˆ– {𝐡})))
6261spcegv 2825 . . . . . 6 ((𝑓 ∘ (inl β†Ύ Ο‰)) ∈ V β†’ ((𝑓 ∘ (inl β†Ύ Ο‰)):ω–1-1β†’(𝐴 βˆ– {𝐡}) β†’ βˆƒπ‘” 𝑔:ω–1-1β†’(𝐴 βˆ– {𝐡})))
6360, 62mpcom 36 . . . . 5 ((𝑓 ∘ (inl β†Ύ Ο‰)):ω–1-1β†’(𝐴 βˆ– {𝐡}) β†’ βˆƒπ‘” 𝑔:ω–1-1β†’(𝐴 βˆ– {𝐡}))
6456, 63syl 14 . . . 4 ((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) β†’ βˆƒπ‘” 𝑔:ω–1-1β†’(𝐴 βˆ– {𝐡}))
65 simpl1 1000 . . . . . . 7 (((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦)
6665adantr 276 . . . . . 6 ((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ Β¬ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦)
67 simpl3 1002 . . . . . . 7 (((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) β†’ 𝐡 ∈ 𝐴)
6867adantr 276 . . . . . 6 ((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ Β¬ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) β†’ 𝐡 ∈ 𝐴)
69 simpr 110 . . . . . . 7 (((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) β†’ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴)
7069adantr 276 . . . . . 6 ((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ Β¬ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) β†’ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴)
71 simpr 110 . . . . . . 7 ((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ Β¬ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) β†’ Β¬ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡)
7271neqned 2354 . . . . . 6 ((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ Β¬ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) β†’ (π‘“β€˜(inrβ€˜βˆ…)) β‰  𝐡)
73 eqid 2177 . . . . . 6 (π‘Ž ∈ Ο‰ ↦ if((π‘“β€˜(inlβ€˜π‘Ž)) = 𝐡, (π‘“β€˜(inrβ€˜βˆ…)), (π‘“β€˜(inlβ€˜π‘Ž)))) = (π‘Ž ∈ Ο‰ ↦ if((π‘“β€˜(inlβ€˜π‘Ž)) = 𝐡, (π‘“β€˜(inrβ€˜βˆ…)), (π‘“β€˜(inlβ€˜π‘Ž))))
7466, 68, 70, 72, 73difinfsnlem 7097 . . . . 5 ((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ Β¬ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) β†’ (π‘Ž ∈ Ο‰ ↦ if((π‘“β€˜(inlβ€˜π‘Ž)) = 𝐡, (π‘“β€˜(inrβ€˜βˆ…)), (π‘“β€˜(inlβ€˜π‘Ž)))):ω–1-1β†’(𝐴 βˆ– {𝐡}))
7558mptex 5742 . . . . . 6 (π‘Ž ∈ Ο‰ ↦ if((π‘“β€˜(inlβ€˜π‘Ž)) = 𝐡, (π‘“β€˜(inrβ€˜βˆ…)), (π‘“β€˜(inlβ€˜π‘Ž)))) ∈ V
76 f1eq1 5416 . . . . . 6 (𝑔 = (π‘Ž ∈ Ο‰ ↦ if((π‘“β€˜(inlβ€˜π‘Ž)) = 𝐡, (π‘“β€˜(inrβ€˜βˆ…)), (π‘“β€˜(inlβ€˜π‘Ž)))) β†’ (𝑔:ω–1-1β†’(𝐴 βˆ– {𝐡}) ↔ (π‘Ž ∈ Ο‰ ↦ if((π‘“β€˜(inlβ€˜π‘Ž)) = 𝐡, (π‘“β€˜(inrβ€˜βˆ…)), (π‘“β€˜(inlβ€˜π‘Ž)))):ω–1-1β†’(𝐴 βˆ– {𝐡})))
7775, 76spcev 2832 . . . . 5 ((π‘Ž ∈ Ο‰ ↦ if((π‘“β€˜(inlβ€˜π‘Ž)) = 𝐡, (π‘“β€˜(inrβ€˜βˆ…)), (π‘“β€˜(inlβ€˜π‘Ž)))):ω–1-1β†’(𝐴 βˆ– {𝐡}) β†’ βˆƒπ‘” 𝑔:ω–1-1β†’(𝐴 βˆ– {𝐡}))
7874, 77syl 14 . . . 4 ((((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) ∧ Β¬ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡) β†’ βˆƒπ‘” 𝑔:ω–1-1β†’(𝐴 βˆ– {𝐡}))
79 f1f 5421 . . . . . . . . 9 (𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴 β†’ 𝑓:(Ο‰ βŠ” 1o)⟢𝐴)
8069, 79syl 14 . . . . . . . 8 (((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) β†’ 𝑓:(Ο‰ βŠ” 1o)⟢𝐴)
8132a1i 9 . . . . . . . 8 (((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) β†’ (inrβ€˜βˆ…) ∈ (Ο‰ βŠ” 1o))
8280, 81ffvelcdmd 5652 . . . . . . 7 (((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) β†’ (π‘“β€˜(inrβ€˜βˆ…)) ∈ 𝐴)
8382, 67jca 306 . . . . . 6 (((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) β†’ ((π‘“β€˜(inrβ€˜βˆ…)) ∈ 𝐴 ∧ 𝐡 ∈ 𝐴))
84 eqeq12 2190 . . . . . . . 8 ((π‘₯ = (π‘“β€˜(inrβ€˜βˆ…)) ∧ 𝑦 = 𝐡) β†’ (π‘₯ = 𝑦 ↔ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡))
8584dcbid 838 . . . . . . 7 ((π‘₯ = (π‘“β€˜(inrβ€˜βˆ…)) ∧ 𝑦 = 𝐡) β†’ (DECID π‘₯ = 𝑦 ↔ DECID (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡))
8685rspc2gv 2853 . . . . . 6 (((π‘“β€˜(inrβ€˜βˆ…)) ∈ 𝐴 ∧ 𝐡 ∈ 𝐴) β†’ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 β†’ DECID (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡))
8783, 65, 86sylc 62 . . . . 5 (((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) β†’ DECID (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡)
88 exmiddc 836 . . . . 5 (DECID (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡 β†’ ((π‘“β€˜(inrβ€˜βˆ…)) = 𝐡 ∨ Β¬ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡))
8987, 88syl 14 . . . 4 (((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) β†’ ((π‘“β€˜(inrβ€˜βˆ…)) = 𝐡 ∨ Β¬ (π‘“β€˜(inrβ€˜βˆ…)) = 𝐡))
9064, 78, 89mpjaodan 798 . . 3 (((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) ∧ 𝑓:(Ο‰ βŠ” 1o)–1-1→𝐴) β†’ βˆƒπ‘” 𝑔:ω–1-1β†’(𝐴 βˆ– {𝐡}))
916, 90exlimddv 1898 . 2 ((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) β†’ βˆƒπ‘” 𝑔:ω–1-1β†’(𝐴 βˆ– {𝐡}))
92 reldom 6744 . . . . . 6 Rel β‰Ό
9392brrelex2i 4670 . . . . 5 (Ο‰ β‰Ό 𝐴 β†’ 𝐴 ∈ V)
94 difexg 4144 . . . . 5 (𝐴 ∈ V β†’ (𝐴 βˆ– {𝐡}) ∈ V)
9593, 94syl 14 . . . 4 (Ο‰ β‰Ό 𝐴 β†’ (𝐴 βˆ– {𝐡}) ∈ V)
96953ad2ant2 1019 . . 3 ((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) β†’ (𝐴 βˆ– {𝐡}) ∈ V)
97 brdomg 6747 . . 3 ((𝐴 βˆ– {𝐡}) ∈ V β†’ (Ο‰ β‰Ό (𝐴 βˆ– {𝐡}) ↔ βˆƒπ‘” 𝑔:ω–1-1β†’(𝐴 βˆ– {𝐡})))
9896, 97syl 14 . 2 ((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) β†’ (Ο‰ β‰Ό (𝐴 βˆ– {𝐡}) ↔ βˆƒπ‘” 𝑔:ω–1-1β†’(𝐴 βˆ– {𝐡})))
9991, 98mpbird 167 1 ((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 DECID π‘₯ = 𝑦 ∧ Ο‰ β‰Ό 𝐴 ∧ 𝐡 ∈ 𝐴) β†’ Ο‰ β‰Ό (𝐴 βˆ– {𝐡}))
Colors of variables: wff set class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708  DECID wdc 834   ∧ w3a 978   = wceq 1353  βˆƒwex 1492   ∈ wcel 2148   β‰  wne 2347  βˆ€wral 2455  Vcvv 2737   βˆ– cdif 3126   βŠ† wss 3129  βˆ…c0 3422  ifcif 3534  {csn 3592   class class class wbr 4003   ↦ cmpt 4064  Ο‰com 4589  ran crn 4627   β†Ύ cres 4628   ∘ ccom 4630   Fn wfn 5211  βŸΆwf 5212  β€“1-1β†’wf1 5213  β€˜cfv 5216  1oc1o 6409   β‰ˆ cen 6737   β‰Ό cdom 6738   βŠ” cdju 7035  inlcinl 7043  inrcinr 7044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-1st 6140  df-2nd 6141  df-1o 6416  df-er 6534  df-en 6740  df-dom 6741  df-dju 7036  df-inl 7045  df-inr 7046  df-case 7082
This theorem is referenced by:  difinfinf  7099
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