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Theorem fin23lem32 10287
Description: Lemma for fin23 10332. Wrap the previous construction into a function to hide the hypotheses. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Hypotheses
Ref Expression
fin23lem.a π‘ˆ = seqΟ‰((𝑖 ∈ Ο‰, 𝑒 ∈ V ↦ if(((π‘‘β€˜π‘–) ∩ 𝑒) = βˆ…, 𝑒, ((π‘‘β€˜π‘–) ∩ 𝑒))), βˆͺ ran 𝑑)
fin23lem17.f 𝐹 = {𝑔 ∣ βˆ€π‘Ž ∈ (𝒫 𝑔 ↑m Ο‰)(βˆ€π‘₯ ∈ Ο‰ (π‘Žβ€˜suc π‘₯) βŠ† (π‘Žβ€˜π‘₯) β†’ ∩ ran π‘Ž ∈ ran π‘Ž)}
fin23lem.b 𝑃 = {𝑣 ∈ Ο‰ ∣ ∩ ran π‘ˆ βŠ† (π‘‘β€˜π‘£)}
fin23lem.c 𝑄 = (𝑀 ∈ Ο‰ ↦ (β„©π‘₯ ∈ 𝑃 (π‘₯ ∩ 𝑃) β‰ˆ 𝑀))
fin23lem.d 𝑅 = (𝑀 ∈ Ο‰ ↦ (β„©π‘₯ ∈ (Ο‰ βˆ– 𝑃)(π‘₯ ∩ (Ο‰ βˆ– 𝑃)) β‰ˆ 𝑀))
fin23lem.e 𝑍 = if(𝑃 ∈ Fin, (𝑑 ∘ 𝑅), ((𝑧 ∈ 𝑃 ↦ ((π‘‘β€˜π‘§) βˆ– ∩ ran π‘ˆ)) ∘ 𝑄))
Assertion
Ref Expression
fin23lem32 (𝐺 ∈ 𝐹 β†’ βˆƒπ‘“βˆ€π‘((𝑏:ω–1-1β†’V ∧ βˆͺ ran 𝑏 βŠ† 𝐺) β†’ ((π‘“β€˜π‘):ω–1-1β†’V ∧ βˆͺ ran (π‘“β€˜π‘) ⊊ βˆͺ ran 𝑏)))
Distinct variable groups:   𝑔,𝑖,𝑑,𝑒,𝑣,π‘₯,𝑧   π‘Ž,𝑏,𝑖,𝑒,𝑑   𝐹,π‘Ž,𝑑   𝑀,π‘Ž,π‘₯,𝑧,𝑃,𝑏   𝑣,π‘Ž,𝑅,𝑏,𝑖,𝑒   π‘ˆ,π‘Ž,𝑏,𝑖,𝑒,𝑣,𝑧   𝑓,π‘Ž,𝑍,𝑏   𝑔,π‘Ž,𝐺,𝑏,𝑑,𝑓,π‘₯
Allowed substitution hints:   𝑃(𝑣,𝑒,𝑑,𝑓,𝑔,𝑖)   𝑄(π‘₯,𝑧,𝑀,𝑣,𝑒,𝑑,𝑓,𝑔,𝑖,π‘Ž,𝑏)   𝑅(π‘₯,𝑧,𝑀,𝑑,𝑓,𝑔)   π‘ˆ(π‘₯,𝑀,𝑑,𝑓,𝑔)   𝐹(π‘₯,𝑧,𝑀,𝑣,𝑒,𝑓,𝑔,𝑖,𝑏)   𝐺(𝑧,𝑀,𝑣,𝑒,𝑖)   𝑍(π‘₯,𝑧,𝑀,𝑣,𝑒,𝑑,𝑔,𝑖)

Proof of Theorem fin23lem32
StepHypRef Expression
1 fin23lem.a . . . . . . . 8 π‘ˆ = seqΟ‰((𝑖 ∈ Ο‰, 𝑒 ∈ V ↦ if(((π‘‘β€˜π‘–) ∩ 𝑒) = βˆ…, 𝑒, ((π‘‘β€˜π‘–) ∩ 𝑒))), βˆͺ ran 𝑑)
2 fin23lem17.f . . . . . . . 8 𝐹 = {𝑔 ∣ βˆ€π‘Ž ∈ (𝒫 𝑔 ↑m Ο‰)(βˆ€π‘₯ ∈ Ο‰ (π‘Žβ€˜suc π‘₯) βŠ† (π‘Žβ€˜π‘₯) β†’ ∩ ran π‘Ž ∈ ran π‘Ž)}
3 fin23lem.b . . . . . . . 8 𝑃 = {𝑣 ∈ Ο‰ ∣ ∩ ran π‘ˆ βŠ† (π‘‘β€˜π‘£)}
4 fin23lem.c . . . . . . . 8 𝑄 = (𝑀 ∈ Ο‰ ↦ (β„©π‘₯ ∈ 𝑃 (π‘₯ ∩ 𝑃) β‰ˆ 𝑀))
5 fin23lem.d . . . . . . . 8 𝑅 = (𝑀 ∈ Ο‰ ↦ (β„©π‘₯ ∈ (Ο‰ βˆ– 𝑃)(π‘₯ ∩ (Ο‰ βˆ– 𝑃)) β‰ˆ 𝑀))
6 fin23lem.e . . . . . . . 8 𝑍 = if(𝑃 ∈ Fin, (𝑑 ∘ 𝑅), ((𝑧 ∈ 𝑃 ↦ ((π‘‘β€˜π‘§) βˆ– ∩ ran π‘ˆ)) ∘ 𝑄))
71, 2, 3, 4, 5, 6fin23lem28 10283 . . . . . . 7 (𝑑:ω–1-1β†’V β†’ 𝑍:ω–1-1β†’V)
87ad2antrl 727 . . . . . 6 ((𝐺 ∈ 𝐹 ∧ (𝑑:ω–1-1β†’V ∧ βˆͺ ran 𝑑 βŠ† 𝐺)) β†’ 𝑍:ω–1-1β†’V)
9 simprl 770 . . . . . . 7 ((𝐺 ∈ 𝐹 ∧ (𝑑:ω–1-1β†’V ∧ βˆͺ ran 𝑑 βŠ† 𝐺)) β†’ 𝑑:ω–1-1β†’V)
10 simpl 484 . . . . . . 7 ((𝐺 ∈ 𝐹 ∧ (𝑑:ω–1-1β†’V ∧ βˆͺ ran 𝑑 βŠ† 𝐺)) β†’ 𝐺 ∈ 𝐹)
11 simprr 772 . . . . . . 7 ((𝐺 ∈ 𝐹 ∧ (𝑑:ω–1-1β†’V ∧ βˆͺ ran 𝑑 βŠ† 𝐺)) β†’ βˆͺ ran 𝑑 βŠ† 𝐺)
121, 2, 3, 4, 5, 6fin23lem31 10286 . . . . . . 7 ((𝑑:ω–1-1β†’V ∧ 𝐺 ∈ 𝐹 ∧ βˆͺ ran 𝑑 βŠ† 𝐺) β†’ βˆͺ ran 𝑍 ⊊ βˆͺ ran 𝑑)
139, 10, 11, 12syl3anc 1372 . . . . . 6 ((𝐺 ∈ 𝐹 ∧ (𝑑:ω–1-1β†’V ∧ βˆͺ ran 𝑑 βŠ† 𝐺)) β†’ βˆͺ ran 𝑍 ⊊ βˆͺ ran 𝑑)
14 f1fn 6744 . . . . . . . . . . . 12 (𝑑:ω–1-1β†’V β†’ 𝑑 Fn Ο‰)
15 dffn3 6686 . . . . . . . . . . . 12 (𝑑 Fn Ο‰ ↔ 𝑑:Ο‰βŸΆran 𝑑)
1614, 15sylib 217 . . . . . . . . . . 11 (𝑑:ω–1-1β†’V β†’ 𝑑:Ο‰βŸΆran 𝑑)
1716ad2antrl 727 . . . . . . . . . 10 ((𝐺 ∈ 𝐹 ∧ (𝑑:ω–1-1β†’V ∧ βˆͺ ran 𝑑 βŠ† 𝐺)) β†’ 𝑑:Ο‰βŸΆran 𝑑)
18 sspwuni 5065 . . . . . . . . . . . 12 (ran 𝑑 βŠ† 𝒫 𝐺 ↔ βˆͺ ran 𝑑 βŠ† 𝐺)
1918biimpri 227 . . . . . . . . . . 11 (βˆͺ ran 𝑑 βŠ† 𝐺 β†’ ran 𝑑 βŠ† 𝒫 𝐺)
2019ad2antll 728 . . . . . . . . . 10 ((𝐺 ∈ 𝐹 ∧ (𝑑:ω–1-1β†’V ∧ βˆͺ ran 𝑑 βŠ† 𝐺)) β†’ ran 𝑑 βŠ† 𝒫 𝐺)
2117, 20fssd 6691 . . . . . . . . 9 ((𝐺 ∈ 𝐹 ∧ (𝑑:ω–1-1β†’V ∧ βˆͺ ran 𝑑 βŠ† 𝐺)) β†’ 𝑑:Ο‰βŸΆπ’« 𝐺)
22 pwexg 5338 . . . . . . . . . . 11 (𝐺 ∈ 𝐹 β†’ 𝒫 𝐺 ∈ V)
2322adantr 482 . . . . . . . . . 10 ((𝐺 ∈ 𝐹 ∧ (𝑑:ω–1-1β†’V ∧ βˆͺ ran 𝑑 βŠ† 𝐺)) β†’ 𝒫 𝐺 ∈ V)
24 vex 3452 . . . . . . . . . . . 12 𝑑 ∈ V
25 f1f 6743 . . . . . . . . . . . 12 (𝑑:ω–1-1β†’V β†’ 𝑑:Ο‰βŸΆV)
26 dmfex 7849 . . . . . . . . . . . 12 ((𝑑 ∈ V ∧ 𝑑:Ο‰βŸΆV) β†’ Ο‰ ∈ V)
2724, 25, 26sylancr 588 . . . . . . . . . . 11 (𝑑:ω–1-1β†’V β†’ Ο‰ ∈ V)
2827ad2antrl 727 . . . . . . . . . 10 ((𝐺 ∈ 𝐹 ∧ (𝑑:ω–1-1β†’V ∧ βˆͺ ran 𝑑 βŠ† 𝐺)) β†’ Ο‰ ∈ V)
2923, 28elmapd 8786 . . . . . . . . 9 ((𝐺 ∈ 𝐹 ∧ (𝑑:ω–1-1β†’V ∧ βˆͺ ran 𝑑 βŠ† 𝐺)) β†’ (𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↔ 𝑑:Ο‰βŸΆπ’« 𝐺))
3021, 29mpbird 257 . . . . . . . 8 ((𝐺 ∈ 𝐹 ∧ (𝑑:ω–1-1β†’V ∧ βˆͺ ran 𝑑 βŠ† 𝐺)) β†’ 𝑑 ∈ (𝒫 𝐺 ↑m Ο‰))
31 f1f 6743 . . . . . . . . . 10 (𝑍:ω–1-1β†’V β†’ 𝑍:Ο‰βŸΆV)
328, 31syl 17 . . . . . . . . 9 ((𝐺 ∈ 𝐹 ∧ (𝑑:ω–1-1β†’V ∧ βˆͺ ran 𝑑 βŠ† 𝐺)) β†’ 𝑍:Ο‰βŸΆV)
3332, 28fexd 7182 . . . . . . . 8 ((𝐺 ∈ 𝐹 ∧ (𝑑:ω–1-1β†’V ∧ βˆͺ ran 𝑑 βŠ† 𝐺)) β†’ 𝑍 ∈ V)
34 eqid 2737 . . . . . . . . 9 (𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍) = (𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)
3534fvmpt2 6964 . . . . . . . 8 ((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ∧ 𝑍 ∈ V) β†’ ((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘) = 𝑍)
3630, 33, 35syl2anc 585 . . . . . . 7 ((𝐺 ∈ 𝐹 ∧ (𝑑:ω–1-1β†’V ∧ βˆͺ ran 𝑑 βŠ† 𝐺)) β†’ ((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘) = 𝑍)
37 f1eq1 6738 . . . . . . . 8 (((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘) = 𝑍 β†’ (((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘):ω–1-1β†’V ↔ 𝑍:ω–1-1β†’V))
38 rneq 5896 . . . . . . . . . 10 (((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘) = 𝑍 β†’ ran ((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘) = ran 𝑍)
3938unieqd 4884 . . . . . . . . 9 (((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘) = 𝑍 β†’ βˆͺ ran ((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘) = βˆͺ ran 𝑍)
4039psseq1d 4057 . . . . . . . 8 (((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘) = 𝑍 β†’ (βˆͺ ran ((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘) ⊊ βˆͺ ran 𝑑 ↔ βˆͺ ran 𝑍 ⊊ βˆͺ ran 𝑑))
4137, 40anbi12d 632 . . . . . . 7 (((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘) = 𝑍 β†’ ((((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘):ω–1-1β†’V ∧ βˆͺ ran ((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘) ⊊ βˆͺ ran 𝑑) ↔ (𝑍:ω–1-1β†’V ∧ βˆͺ ran 𝑍 ⊊ βˆͺ ran 𝑑)))
4236, 41syl 17 . . . . . 6 ((𝐺 ∈ 𝐹 ∧ (𝑑:ω–1-1β†’V ∧ βˆͺ ran 𝑑 βŠ† 𝐺)) β†’ ((((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘):ω–1-1β†’V ∧ βˆͺ ran ((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘) ⊊ βˆͺ ran 𝑑) ↔ (𝑍:ω–1-1β†’V ∧ βˆͺ ran 𝑍 ⊊ βˆͺ ran 𝑑)))
438, 13, 42mpbir2and 712 . . . . 5 ((𝐺 ∈ 𝐹 ∧ (𝑑:ω–1-1β†’V ∧ βˆͺ ran 𝑑 βŠ† 𝐺)) β†’ (((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘):ω–1-1β†’V ∧ βˆͺ ran ((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘) ⊊ βˆͺ ran 𝑑))
4443ex 414 . . . 4 (𝐺 ∈ 𝐹 β†’ ((𝑑:ω–1-1β†’V ∧ βˆͺ ran 𝑑 βŠ† 𝐺) β†’ (((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘):ω–1-1β†’V ∧ βˆͺ ran ((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘) ⊊ βˆͺ ran 𝑑)))
4544alrimiv 1931 . . 3 (𝐺 ∈ 𝐹 β†’ βˆ€π‘‘((𝑑:ω–1-1β†’V ∧ βˆͺ ran 𝑑 βŠ† 𝐺) β†’ (((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘):ω–1-1β†’V ∧ βˆͺ ran ((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘) ⊊ βˆͺ ran 𝑑)))
46 ovex 7395 . . . . 5 (𝒫 𝐺 ↑m Ο‰) ∈ V
4746mptex 7178 . . . 4 (𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍) ∈ V
48 nfmpt1 5218 . . . . . 6 Ⅎ𝑑(𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)
4948nfeq2 2925 . . . . 5 Ⅎ𝑑 𝑓 = (𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)
50 fveq1 6846 . . . . . . . 8 (𝑓 = (𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍) β†’ (π‘“β€˜π‘‘) = ((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘))
51 f1eq1 6738 . . . . . . . 8 ((π‘“β€˜π‘‘) = ((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘) β†’ ((π‘“β€˜π‘‘):ω–1-1β†’V ↔ ((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘):ω–1-1β†’V))
5250, 51syl 17 . . . . . . 7 (𝑓 = (𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍) β†’ ((π‘“β€˜π‘‘):ω–1-1β†’V ↔ ((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘):ω–1-1β†’V))
5350rneqd 5898 . . . . . . . . 9 (𝑓 = (𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍) β†’ ran (π‘“β€˜π‘‘) = ran ((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘))
5453unieqd 4884 . . . . . . . 8 (𝑓 = (𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍) β†’ βˆͺ ran (π‘“β€˜π‘‘) = βˆͺ ran ((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘))
5554psseq1d 4057 . . . . . . 7 (𝑓 = (𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍) β†’ (βˆͺ ran (π‘“β€˜π‘‘) ⊊ βˆͺ ran 𝑑 ↔ βˆͺ ran ((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘) ⊊ βˆͺ ran 𝑑))
5652, 55anbi12d 632 . . . . . 6 (𝑓 = (𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍) β†’ (((π‘“β€˜π‘‘):ω–1-1β†’V ∧ βˆͺ ran (π‘“β€˜π‘‘) ⊊ βˆͺ ran 𝑑) ↔ (((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘):ω–1-1β†’V ∧ βˆͺ ran ((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘) ⊊ βˆͺ ran 𝑑)))
5756imbi2d 341 . . . . 5 (𝑓 = (𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍) β†’ (((𝑑:ω–1-1β†’V ∧ βˆͺ ran 𝑑 βŠ† 𝐺) β†’ ((π‘“β€˜π‘‘):ω–1-1β†’V ∧ βˆͺ ran (π‘“β€˜π‘‘) ⊊ βˆͺ ran 𝑑)) ↔ ((𝑑:ω–1-1β†’V ∧ βˆͺ ran 𝑑 βŠ† 𝐺) β†’ (((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘):ω–1-1β†’V ∧ βˆͺ ran ((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘) ⊊ βˆͺ ran 𝑑))))
5849, 57albid 2216 . . . 4 (𝑓 = (𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍) β†’ (βˆ€π‘‘((𝑑:ω–1-1β†’V ∧ βˆͺ ran 𝑑 βŠ† 𝐺) β†’ ((π‘“β€˜π‘‘):ω–1-1β†’V ∧ βˆͺ ran (π‘“β€˜π‘‘) ⊊ βˆͺ ran 𝑑)) ↔ βˆ€π‘‘((𝑑:ω–1-1β†’V ∧ βˆͺ ran 𝑑 βŠ† 𝐺) β†’ (((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘):ω–1-1β†’V ∧ βˆͺ ran ((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘) ⊊ βˆͺ ran 𝑑))))
5947, 58spcev 3568 . . 3 (βˆ€π‘‘((𝑑:ω–1-1β†’V ∧ βˆͺ ran 𝑑 βŠ† 𝐺) β†’ (((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘):ω–1-1β†’V ∧ βˆͺ ran ((𝑑 ∈ (𝒫 𝐺 ↑m Ο‰) ↦ 𝑍)β€˜π‘‘) ⊊ βˆͺ ran 𝑑)) β†’ βˆƒπ‘“βˆ€π‘‘((𝑑:ω–1-1β†’V ∧ βˆͺ ran 𝑑 βŠ† 𝐺) β†’ ((π‘“β€˜π‘‘):ω–1-1β†’V ∧ βˆͺ ran (π‘“β€˜π‘‘) ⊊ βˆͺ ran 𝑑)))
6045, 59syl 17 . 2 (𝐺 ∈ 𝐹 β†’ βˆƒπ‘“βˆ€π‘‘((𝑑:ω–1-1β†’V ∧ βˆͺ ran 𝑑 βŠ† 𝐺) β†’ ((π‘“β€˜π‘‘):ω–1-1β†’V ∧ βˆͺ ran (π‘“β€˜π‘‘) ⊊ βˆͺ ran 𝑑)))
61 f1eq1 6738 . . . . . 6 (𝑏 = 𝑑 β†’ (𝑏:ω–1-1β†’V ↔ 𝑑:ω–1-1β†’V))
62 rneq 5896 . . . . . . . 8 (𝑏 = 𝑑 β†’ ran 𝑏 = ran 𝑑)
6362unieqd 4884 . . . . . . 7 (𝑏 = 𝑑 β†’ βˆͺ ran 𝑏 = βˆͺ ran 𝑑)
6463sseq1d 3980 . . . . . 6 (𝑏 = 𝑑 β†’ (βˆͺ ran 𝑏 βŠ† 𝐺 ↔ βˆͺ ran 𝑑 βŠ† 𝐺))
6561, 64anbi12d 632 . . . . 5 (𝑏 = 𝑑 β†’ ((𝑏:ω–1-1β†’V ∧ βˆͺ ran 𝑏 βŠ† 𝐺) ↔ (𝑑:ω–1-1β†’V ∧ βˆͺ ran 𝑑 βŠ† 𝐺)))
66 fveq2 6847 . . . . . . 7 (𝑏 = 𝑑 β†’ (π‘“β€˜π‘) = (π‘“β€˜π‘‘))
67 f1eq1 6738 . . . . . . 7 ((π‘“β€˜π‘) = (π‘“β€˜π‘‘) β†’ ((π‘“β€˜π‘):ω–1-1β†’V ↔ (π‘“β€˜π‘‘):ω–1-1β†’V))
6866, 67syl 17 . . . . . 6 (𝑏 = 𝑑 β†’ ((π‘“β€˜π‘):ω–1-1β†’V ↔ (π‘“β€˜π‘‘):ω–1-1β†’V))
6966rneqd 5898 . . . . . . . 8 (𝑏 = 𝑑 β†’ ran (π‘“β€˜π‘) = ran (π‘“β€˜π‘‘))
7069unieqd 4884 . . . . . . 7 (𝑏 = 𝑑 β†’ βˆͺ ran (π‘“β€˜π‘) = βˆͺ ran (π‘“β€˜π‘‘))
7170, 63psseq12d 4059 . . . . . 6 (𝑏 = 𝑑 β†’ (βˆͺ ran (π‘“β€˜π‘) ⊊ βˆͺ ran 𝑏 ↔ βˆͺ ran (π‘“β€˜π‘‘) ⊊ βˆͺ ran 𝑑))
7268, 71anbi12d 632 . . . . 5 (𝑏 = 𝑑 β†’ (((π‘“β€˜π‘):ω–1-1β†’V ∧ βˆͺ ran (π‘“β€˜π‘) ⊊ βˆͺ ran 𝑏) ↔ ((π‘“β€˜π‘‘):ω–1-1β†’V ∧ βˆͺ ran (π‘“β€˜π‘‘) ⊊ βˆͺ ran 𝑑)))
7365, 72imbi12d 345 . . . 4 (𝑏 = 𝑑 β†’ (((𝑏:ω–1-1β†’V ∧ βˆͺ ran 𝑏 βŠ† 𝐺) β†’ ((π‘“β€˜π‘):ω–1-1β†’V ∧ βˆͺ ran (π‘“β€˜π‘) ⊊ βˆͺ ran 𝑏)) ↔ ((𝑑:ω–1-1β†’V ∧ βˆͺ ran 𝑑 βŠ† 𝐺) β†’ ((π‘“β€˜π‘‘):ω–1-1β†’V ∧ βˆͺ ran (π‘“β€˜π‘‘) ⊊ βˆͺ ran 𝑑))))
7473cbvalvw 2040 . . 3 (βˆ€π‘((𝑏:ω–1-1β†’V ∧ βˆͺ ran 𝑏 βŠ† 𝐺) β†’ ((π‘“β€˜π‘):ω–1-1β†’V ∧ βˆͺ ran (π‘“β€˜π‘) ⊊ βˆͺ ran 𝑏)) ↔ βˆ€π‘‘((𝑑:ω–1-1β†’V ∧ βˆͺ ran 𝑑 βŠ† 𝐺) β†’ ((π‘“β€˜π‘‘):ω–1-1β†’V ∧ βˆͺ ran (π‘“β€˜π‘‘) ⊊ βˆͺ ran 𝑑)))
7574exbii 1851 . 2 (βˆƒπ‘“βˆ€π‘((𝑏:ω–1-1β†’V ∧ βˆͺ ran 𝑏 βŠ† 𝐺) β†’ ((π‘“β€˜π‘):ω–1-1β†’V ∧ βˆͺ ran (π‘“β€˜π‘) ⊊ βˆͺ ran 𝑏)) ↔ βˆƒπ‘“βˆ€π‘‘((𝑑:ω–1-1β†’V ∧ βˆͺ ran 𝑑 βŠ† 𝐺) β†’ ((π‘“β€˜π‘‘):ω–1-1β†’V ∧ βˆͺ ran (π‘“β€˜π‘‘) ⊊ βˆͺ ran 𝑑)))
7660, 75sylibr 233 1 (𝐺 ∈ 𝐹 β†’ βˆƒπ‘“βˆ€π‘((𝑏:ω–1-1β†’V ∧ βˆͺ ran 𝑏 βŠ† 𝐺) β†’ ((π‘“β€˜π‘):ω–1-1β†’V ∧ βˆͺ ran (π‘“β€˜π‘) ⊊ βˆͺ ran 𝑏)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397  βˆ€wal 1540   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {cab 2714  βˆ€wral 3065  {crab 3410  Vcvv 3448   βˆ– cdif 3912   ∩ cin 3914   βŠ† wss 3915   ⊊ wpss 3916  βˆ…c0 4287  ifcif 4491  π’« cpw 4565  βˆͺ cuni 4870  βˆ© cint 4912   class class class wbr 5110   ↦ cmpt 5193  ran crn 5639   ∘ ccom 5642  suc csuc 6324   Fn wfn 6496  βŸΆwf 6497  β€“1-1β†’wf1 6498  β€˜cfv 6501  β„©crio 7317  (class class class)co 7362   ∈ cmpo 7364  Ο‰com 7807  seqΟ‰cseqom 8398   ↑m cmap 8772   β‰ˆ cen 8887  Fincfn 8890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-seqom 8399  df-1o 8417  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-card 9882
This theorem is referenced by:  fin23lem33  10288
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