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Theorem enfixsn 9120
Description: Given two equipollent sets, a bijection can always be chosen which fixes a single point. (Contributed by Stefan O'Rear, 9-Jul-2015.)
Assertion
Ref Expression
enfixsn ((𝐴𝑋𝐵𝑌𝑋𝑌) → ∃𝑓(𝑓:𝑋1-1-onto𝑌 ∧ (𝑓𝐴) = 𝐵))
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓   𝑓,𝑋   𝑓,𝑌

Proof of Theorem enfixsn
Dummy variables 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 1137 . . 3 ((𝐴𝑋𝐵𝑌𝑋𝑌) → 𝑋𝑌)
2 bren 8994 . . 3 (𝑋𝑌 ↔ ∃𝑔 𝑔:𝑋1-1-onto𝑌)
31, 2sylib 218 . 2 ((𝐴𝑋𝐵𝑌𝑋𝑌) → ∃𝑔 𝑔:𝑋1-1-onto𝑌)
4 relen 8989 . . . . . . . 8 Rel ≈
54brrelex2i 5746 . . . . . . 7 (𝑋𝑌𝑌 ∈ V)
653ad2ant3 1134 . . . . . 6 ((𝐴𝑋𝐵𝑌𝑋𝑌) → 𝑌 ∈ V)
76adantr 480 . . . . 5 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ 𝑔:𝑋1-1-onto𝑌) → 𝑌 ∈ V)
8 f1of 6849 . . . . . . 7 (𝑔:𝑋1-1-onto𝑌𝑔:𝑋𝑌)
98adantl 481 . . . . . 6 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ 𝑔:𝑋1-1-onto𝑌) → 𝑔:𝑋𝑌)
10 simpl1 1190 . . . . . 6 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ 𝑔:𝑋1-1-onto𝑌) → 𝐴𝑋)
119, 10ffvelcdmd 7105 . . . . 5 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ 𝑔:𝑋1-1-onto𝑌) → (𝑔𝐴) ∈ 𝑌)
12 simpl2 1191 . . . . 5 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ 𝑔:𝑋1-1-onto𝑌) → 𝐵𝑌)
13 difsnen 9092 . . . . 5 ((𝑌 ∈ V ∧ (𝑔𝐴) ∈ 𝑌𝐵𝑌) → (𝑌 ∖ {(𝑔𝐴)}) ≈ (𝑌 ∖ {𝐵}))
147, 11, 12, 13syl3anc 1370 . . . 4 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ 𝑔:𝑋1-1-onto𝑌) → (𝑌 ∖ {(𝑔𝐴)}) ≈ (𝑌 ∖ {𝐵}))
15 bren 8994 . . . 4 ((𝑌 ∖ {(𝑔𝐴)}) ≈ (𝑌 ∖ {𝐵}) ↔ ∃ :(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))
1614, 15sylib 218 . . 3 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ 𝑔:𝑋1-1-onto𝑌) → ∃ :(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))
17 fvex 6920 . . . . . . . . . . 11 (𝑔𝐴) ∈ V
1817a1i 11 . . . . . . . . . 10 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ (𝑔:𝑋1-1-onto𝑌:(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → (𝑔𝐴) ∈ V)
19 simpl2 1191 . . . . . . . . . 10 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ (𝑔:𝑋1-1-onto𝑌:(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → 𝐵𝑌)
20 f1osng 6890 . . . . . . . . . 10 (((𝑔𝐴) ∈ V ∧ 𝐵𝑌) → {⟨(𝑔𝐴), 𝐵⟩}:{(𝑔𝐴)}–1-1-onto→{𝐵})
2118, 19, 20syl2anc 584 . . . . . . . . 9 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ (𝑔:𝑋1-1-onto𝑌:(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → {⟨(𝑔𝐴), 𝐵⟩}:{(𝑔𝐴)}–1-1-onto→{𝐵})
22 simprr 773 . . . . . . . . 9 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ (𝑔:𝑋1-1-onto𝑌:(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → :(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))
23 disjdif 4478 . . . . . . . . . 10 ({(𝑔𝐴)} ∩ (𝑌 ∖ {(𝑔𝐴)})) = ∅
2423a1i 11 . . . . . . . . 9 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ (𝑔:𝑋1-1-onto𝑌:(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → ({(𝑔𝐴)} ∩ (𝑌 ∖ {(𝑔𝐴)})) = ∅)
25 disjdif 4478 . . . . . . . . . 10 ({𝐵} ∩ (𝑌 ∖ {𝐵})) = ∅
2625a1i 11 . . . . . . . . 9 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ (𝑔:𝑋1-1-onto𝑌:(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → ({𝐵} ∩ (𝑌 ∖ {𝐵})) = ∅)
27 f1oun 6868 . . . . . . . . 9 ((({⟨(𝑔𝐴), 𝐵⟩}:{(𝑔𝐴)}–1-1-onto→{𝐵} ∧ :(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵})) ∧ (({(𝑔𝐴)} ∩ (𝑌 ∖ {(𝑔𝐴)})) = ∅ ∧ ({𝐵} ∩ (𝑌 ∖ {𝐵})) = ∅)) → ({⟨(𝑔𝐴), 𝐵⟩} ∪ ):({(𝑔𝐴)} ∪ (𝑌 ∖ {(𝑔𝐴)}))–1-1-onto→({𝐵} ∪ (𝑌 ∖ {𝐵})))
2821, 22, 24, 26, 27syl22anc 839 . . . . . . . 8 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ (𝑔:𝑋1-1-onto𝑌:(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → ({⟨(𝑔𝐴), 𝐵⟩} ∪ ):({(𝑔𝐴)} ∪ (𝑌 ∖ {(𝑔𝐴)}))–1-1-onto→({𝐵} ∪ (𝑌 ∖ {𝐵})))
298ad2antrl 728 . . . . . . . . . . 11 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ (𝑔:𝑋1-1-onto𝑌:(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → 𝑔:𝑋𝑌)
30 simpl1 1190 . . . . . . . . . . 11 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ (𝑔:𝑋1-1-onto𝑌:(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → 𝐴𝑋)
3129, 30ffvelcdmd 7105 . . . . . . . . . 10 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ (𝑔:𝑋1-1-onto𝑌:(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → (𝑔𝐴) ∈ 𝑌)
32 uncom 4168 . . . . . . . . . . 11 ({(𝑔𝐴)} ∪ (𝑌 ∖ {(𝑔𝐴)})) = ((𝑌 ∖ {(𝑔𝐴)}) ∪ {(𝑔𝐴)})
33 difsnid 4815 . . . . . . . . . . 11 ((𝑔𝐴) ∈ 𝑌 → ((𝑌 ∖ {(𝑔𝐴)}) ∪ {(𝑔𝐴)}) = 𝑌)
3432, 33eqtrid 2787 . . . . . . . . . 10 ((𝑔𝐴) ∈ 𝑌 → ({(𝑔𝐴)} ∪ (𝑌 ∖ {(𝑔𝐴)})) = 𝑌)
3531, 34syl 17 . . . . . . . . 9 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ (𝑔:𝑋1-1-onto𝑌:(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → ({(𝑔𝐴)} ∪ (𝑌 ∖ {(𝑔𝐴)})) = 𝑌)
36 uncom 4168 . . . . . . . . . . 11 ({𝐵} ∪ (𝑌 ∖ {𝐵})) = ((𝑌 ∖ {𝐵}) ∪ {𝐵})
37 difsnid 4815 . . . . . . . . . . 11 (𝐵𝑌 → ((𝑌 ∖ {𝐵}) ∪ {𝐵}) = 𝑌)
3836, 37eqtrid 2787 . . . . . . . . . 10 (𝐵𝑌 → ({𝐵} ∪ (𝑌 ∖ {𝐵})) = 𝑌)
3919, 38syl 17 . . . . . . . . 9 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ (𝑔:𝑋1-1-onto𝑌:(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → ({𝐵} ∪ (𝑌 ∖ {𝐵})) = 𝑌)
40 f1oeq23 6840 . . . . . . . . 9 ((({(𝑔𝐴)} ∪ (𝑌 ∖ {(𝑔𝐴)})) = 𝑌 ∧ ({𝐵} ∪ (𝑌 ∖ {𝐵})) = 𝑌) → (({⟨(𝑔𝐴), 𝐵⟩} ∪ ):({(𝑔𝐴)} ∪ (𝑌 ∖ {(𝑔𝐴)}))–1-1-onto→({𝐵} ∪ (𝑌 ∖ {𝐵})) ↔ ({⟨(𝑔𝐴), 𝐵⟩} ∪ ):𝑌1-1-onto𝑌))
4135, 39, 40syl2anc 584 . . . . . . . 8 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ (𝑔:𝑋1-1-onto𝑌:(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → (({⟨(𝑔𝐴), 𝐵⟩} ∪ ):({(𝑔𝐴)} ∪ (𝑌 ∖ {(𝑔𝐴)}))–1-1-onto→({𝐵} ∪ (𝑌 ∖ {𝐵})) ↔ ({⟨(𝑔𝐴), 𝐵⟩} ∪ ):𝑌1-1-onto𝑌))
4228, 41mpbid 232 . . . . . . 7 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ (𝑔:𝑋1-1-onto𝑌:(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → ({⟨(𝑔𝐴), 𝐵⟩} ∪ ):𝑌1-1-onto𝑌)
43 simprl 771 . . . . . . 7 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ (𝑔:𝑋1-1-onto𝑌:(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → 𝑔:𝑋1-1-onto𝑌)
44 f1oco 6872 . . . . . . 7 ((({⟨(𝑔𝐴), 𝐵⟩} ∪ ):𝑌1-1-onto𝑌𝑔:𝑋1-1-onto𝑌) → (({⟨(𝑔𝐴), 𝐵⟩} ∪ ) ∘ 𝑔):𝑋1-1-onto𝑌)
4542, 43, 44syl2anc 584 . . . . . 6 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ (𝑔:𝑋1-1-onto𝑌:(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → (({⟨(𝑔𝐴), 𝐵⟩} ∪ ) ∘ 𝑔):𝑋1-1-onto𝑌)
46 f1ofn 6850 . . . . . . . . 9 (𝑔:𝑋1-1-onto𝑌𝑔 Fn 𝑋)
4746ad2antrl 728 . . . . . . . 8 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ (𝑔:𝑋1-1-onto𝑌:(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → 𝑔 Fn 𝑋)
48 fvco2 7006 . . . . . . . 8 ((𝑔 Fn 𝑋𝐴𝑋) → ((({⟨(𝑔𝐴), 𝐵⟩} ∪ ) ∘ 𝑔)‘𝐴) = (({⟨(𝑔𝐴), 𝐵⟩} ∪ )‘(𝑔𝐴)))
4947, 30, 48syl2anc 584 . . . . . . 7 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ (𝑔:𝑋1-1-onto𝑌:(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → ((({⟨(𝑔𝐴), 𝐵⟩} ∪ ) ∘ 𝑔)‘𝐴) = (({⟨(𝑔𝐴), 𝐵⟩} ∪ )‘(𝑔𝐴)))
50 f1ofn 6850 . . . . . . . . 9 ({⟨(𝑔𝐴), 𝐵⟩}:{(𝑔𝐴)}–1-1-onto→{𝐵} → {⟨(𝑔𝐴), 𝐵⟩} Fn {(𝑔𝐴)})
5121, 50syl 17 . . . . . . . 8 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ (𝑔:𝑋1-1-onto𝑌:(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → {⟨(𝑔𝐴), 𝐵⟩} Fn {(𝑔𝐴)})
52 f1ofn 6850 . . . . . . . . 9 (:(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}) → Fn (𝑌 ∖ {(𝑔𝐴)}))
5352ad2antll 729 . . . . . . . 8 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ (𝑔:𝑋1-1-onto𝑌:(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → Fn (𝑌 ∖ {(𝑔𝐴)}))
5417snid 4667 . . . . . . . . 9 (𝑔𝐴) ∈ {(𝑔𝐴)}
5554a1i 11 . . . . . . . 8 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ (𝑔:𝑋1-1-onto𝑌:(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → (𝑔𝐴) ∈ {(𝑔𝐴)})
56 fvun1 7000 . . . . . . . 8 (({⟨(𝑔𝐴), 𝐵⟩} Fn {(𝑔𝐴)} ∧ Fn (𝑌 ∖ {(𝑔𝐴)}) ∧ (({(𝑔𝐴)} ∩ (𝑌 ∖ {(𝑔𝐴)})) = ∅ ∧ (𝑔𝐴) ∈ {(𝑔𝐴)})) → (({⟨(𝑔𝐴), 𝐵⟩} ∪ )‘(𝑔𝐴)) = ({⟨(𝑔𝐴), 𝐵⟩}‘(𝑔𝐴)))
5751, 53, 24, 55, 56syl112anc 1373 . . . . . . 7 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ (𝑔:𝑋1-1-onto𝑌:(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → (({⟨(𝑔𝐴), 𝐵⟩} ∪ )‘(𝑔𝐴)) = ({⟨(𝑔𝐴), 𝐵⟩}‘(𝑔𝐴)))
58 fvsng 7200 . . . . . . . 8 (((𝑔𝐴) ∈ V ∧ 𝐵𝑌) → ({⟨(𝑔𝐴), 𝐵⟩}‘(𝑔𝐴)) = 𝐵)
5918, 19, 58syl2anc 584 . . . . . . 7 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ (𝑔:𝑋1-1-onto𝑌:(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → ({⟨(𝑔𝐴), 𝐵⟩}‘(𝑔𝐴)) = 𝐵)
6049, 57, 593eqtrd 2779 . . . . . 6 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ (𝑔:𝑋1-1-onto𝑌:(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → ((({⟨(𝑔𝐴), 𝐵⟩} ∪ ) ∘ 𝑔)‘𝐴) = 𝐵)
61 snex 5442 . . . . . . . . 9 {⟨(𝑔𝐴), 𝐵⟩} ∈ V
62 vex 3482 . . . . . . . . 9 ∈ V
6361, 62unex 7763 . . . . . . . 8 ({⟨(𝑔𝐴), 𝐵⟩} ∪ ) ∈ V
64 vex 3482 . . . . . . . 8 𝑔 ∈ V
6563, 64coex 7953 . . . . . . 7 (({⟨(𝑔𝐴), 𝐵⟩} ∪ ) ∘ 𝑔) ∈ V
66 f1oeq1 6837 . . . . . . . 8 (𝑓 = (({⟨(𝑔𝐴), 𝐵⟩} ∪ ) ∘ 𝑔) → (𝑓:𝑋1-1-onto𝑌 ↔ (({⟨(𝑔𝐴), 𝐵⟩} ∪ ) ∘ 𝑔):𝑋1-1-onto𝑌))
67 fveq1 6906 . . . . . . . . 9 (𝑓 = (({⟨(𝑔𝐴), 𝐵⟩} ∪ ) ∘ 𝑔) → (𝑓𝐴) = ((({⟨(𝑔𝐴), 𝐵⟩} ∪ ) ∘ 𝑔)‘𝐴))
6867eqeq1d 2737 . . . . . . . 8 (𝑓 = (({⟨(𝑔𝐴), 𝐵⟩} ∪ ) ∘ 𝑔) → ((𝑓𝐴) = 𝐵 ↔ ((({⟨(𝑔𝐴), 𝐵⟩} ∪ ) ∘ 𝑔)‘𝐴) = 𝐵))
6966, 68anbi12d 632 . . . . . . 7 (𝑓 = (({⟨(𝑔𝐴), 𝐵⟩} ∪ ) ∘ 𝑔) → ((𝑓:𝑋1-1-onto𝑌 ∧ (𝑓𝐴) = 𝐵) ↔ ((({⟨(𝑔𝐴), 𝐵⟩} ∪ ) ∘ 𝑔):𝑋1-1-onto𝑌 ∧ ((({⟨(𝑔𝐴), 𝐵⟩} ∪ ) ∘ 𝑔)‘𝐴) = 𝐵)))
7065, 69spcev 3606 . . . . . 6 (((({⟨(𝑔𝐴), 𝐵⟩} ∪ ) ∘ 𝑔):𝑋1-1-onto𝑌 ∧ ((({⟨(𝑔𝐴), 𝐵⟩} ∪ ) ∘ 𝑔)‘𝐴) = 𝐵) → ∃𝑓(𝑓:𝑋1-1-onto𝑌 ∧ (𝑓𝐴) = 𝐵))
7145, 60, 70syl2anc 584 . . . . 5 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ (𝑔:𝑋1-1-onto𝑌:(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}))) → ∃𝑓(𝑓:𝑋1-1-onto𝑌 ∧ (𝑓𝐴) = 𝐵))
7271expr 456 . . . 4 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ 𝑔:𝑋1-1-onto𝑌) → (:(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}) → ∃𝑓(𝑓:𝑋1-1-onto𝑌 ∧ (𝑓𝐴) = 𝐵)))
7372exlimdv 1931 . . 3 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ 𝑔:𝑋1-1-onto𝑌) → (∃ :(𝑌 ∖ {(𝑔𝐴)})–1-1-onto→(𝑌 ∖ {𝐵}) → ∃𝑓(𝑓:𝑋1-1-onto𝑌 ∧ (𝑓𝐴) = 𝐵)))
7416, 73mpd 15 . 2 (((𝐴𝑋𝐵𝑌𝑋𝑌) ∧ 𝑔:𝑋1-1-onto𝑌) → ∃𝑓(𝑓:𝑋1-1-onto𝑌 ∧ (𝑓𝐴) = 𝐵))
753, 74exlimddv 1933 1 ((𝐴𝑋𝐵𝑌𝑋𝑌) → ∃𝑓(𝑓:𝑋1-1-onto𝑌 ∧ (𝑓𝐴) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  Vcvv 3478  cdif 3960  cun 3961  cin 3962  c0 4339  {csn 4631  cop 4637   class class class wbr 5148  ccom 5693   Fn wfn 6558  wf 6559  1-1-ontowf1o 6562  cfv 6563  cen 8981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-en 8985
This theorem is referenced by:  mapfien2  9447
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