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Theorem tz7.49c 7486
Description: Corollary of Proposition 7.49 of [TakeutiZaring] p. 51. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 19-Jan-2013.)
Hypothesis
Ref Expression
tz7.49c.1 𝐹 Fn On
Assertion
Ref Expression
tz7.49c ((𝐴𝐵 ∧ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))) → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem tz7.49c
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 tz7.49c.1 . . 3 𝐹 Fn On
2 biid 251 . . 3 (∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) ↔ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
31, 2tz7.49 7485 . 2 ((𝐴𝐵 ∧ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))) → ∃𝑥 ∈ On (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)))
4 3simpc 1058 . . . 4 ((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)) → ((𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)))
5 onss 6937 . . . . . . . . 9 (𝑥 ∈ On → 𝑥 ⊆ On)
6 fnssres 5962 . . . . . . . . 9 ((𝐹 Fn On ∧ 𝑥 ⊆ On) → (𝐹𝑥) Fn 𝑥)
71, 5, 6sylancr 694 . . . . . . . 8 (𝑥 ∈ On → (𝐹𝑥) Fn 𝑥)
8 df-ima 5087 . . . . . . . . . 10 (𝐹𝑥) = ran (𝐹𝑥)
98eqeq1i 2626 . . . . . . . . 9 ((𝐹𝑥) = 𝐴 ↔ ran (𝐹𝑥) = 𝐴)
109biimpi 206 . . . . . . . 8 ((𝐹𝑥) = 𝐴 → ran (𝐹𝑥) = 𝐴)
117, 10anim12i 589 . . . . . . 7 ((𝑥 ∈ On ∧ (𝐹𝑥) = 𝐴) → ((𝐹𝑥) Fn 𝑥 ∧ ran (𝐹𝑥) = 𝐴))
1211anim1i 591 . . . . . 6 (((𝑥 ∈ On ∧ (𝐹𝑥) = 𝐴) ∧ Fun (𝐹𝑥)) → (((𝐹𝑥) Fn 𝑥 ∧ ran (𝐹𝑥) = 𝐴) ∧ Fun (𝐹𝑥)))
13 dff1o2 6099 . . . . . . 7 ((𝐹𝑥):𝑥1-1-onto𝐴 ↔ ((𝐹𝑥) Fn 𝑥 ∧ Fun (𝐹𝑥) ∧ ran (𝐹𝑥) = 𝐴))
14 3anan32 1048 . . . . . . 7 (((𝐹𝑥) Fn 𝑥 ∧ Fun (𝐹𝑥) ∧ ran (𝐹𝑥) = 𝐴) ↔ (((𝐹𝑥) Fn 𝑥 ∧ ran (𝐹𝑥) = 𝐴) ∧ Fun (𝐹𝑥)))
1513, 14bitri 264 . . . . . 6 ((𝐹𝑥):𝑥1-1-onto𝐴 ↔ (((𝐹𝑥) Fn 𝑥 ∧ ran (𝐹𝑥) = 𝐴) ∧ Fun (𝐹𝑥)))
1612, 15sylibr 224 . . . . 5 (((𝑥 ∈ On ∧ (𝐹𝑥) = 𝐴) ∧ Fun (𝐹𝑥)) → (𝐹𝑥):𝑥1-1-onto𝐴)
1716expl 647 . . . 4 (𝑥 ∈ On → (((𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)) → (𝐹𝑥):𝑥1-1-onto𝐴))
184, 17syl5 34 . . 3 (𝑥 ∈ On → ((∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)) → (𝐹𝑥):𝑥1-1-onto𝐴))
1918reximia 3003 . 2 (∃𝑥 ∈ On (∀𝑦𝑥 (𝐴 ∖ (𝐹𝑦)) ≠ ∅ ∧ (𝐹𝑥) = 𝐴 ∧ Fun (𝐹𝑥)) → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴)
203, 19syl 17 1 ((𝐴𝐵 ∧ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))) → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  cdif 3552  wss 3555  c0 3891  ccnv 5073  ran crn 5075  cres 5076  cima 5077  Oncon0 5682  Fun wfun 5841   Fn wfn 5842  1-1-ontowf1o 5846  cfv 5847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855
This theorem is referenced by:  dfac8alem  8796  dnnumch1  37094
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