MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tz7.48-1 Structured version   Visualization version   GIF version

Theorem tz7.48-1 7523
Description: Proposition 7.48(1) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.)
Hypothesis
Ref Expression
tz7.48.1 𝐹 Fn On
Assertion
Ref Expression
tz7.48-1 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ran 𝐹𝐴)
Distinct variable groups:   𝑥,𝐹   𝑥,𝐴

Proof of Theorem tz7.48-1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 vex 3198 . . . . 5 𝑦 ∈ V
21elrn2 5354 . . . 4 (𝑦 ∈ ran 𝐹 ↔ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐹)
3 vex 3198 . . . . . . . . 9 𝑥 ∈ V
43, 1opeldm 5317 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥 ∈ dom 𝐹)
5 tz7.48.1 . . . . . . . . 9 𝐹 Fn On
6 fndm 5978 . . . . . . . . 9 (𝐹 Fn On → dom 𝐹 = On)
75, 6ax-mp 5 . . . . . . . 8 dom 𝐹 = On
84, 7syl6eleq 2709 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑥 ∈ On)
98ancri 574 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ On ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
10 fnopfvb 6224 . . . . . . . 8 ((𝐹 Fn On ∧ 𝑥 ∈ On) → ((𝐹𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
115, 10mpan 705 . . . . . . 7 (𝑥 ∈ On → ((𝐹𝑥) = 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
1211pm5.32i 668 . . . . . 6 ((𝑥 ∈ On ∧ (𝐹𝑥) = 𝑦) ↔ (𝑥 ∈ On ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹))
139, 12sylibr 224 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐹 → (𝑥 ∈ On ∧ (𝐹𝑥) = 𝑦))
1413eximi 1760 . . . 4 (∃𝑥𝑥, 𝑦⟩ ∈ 𝐹 → ∃𝑥(𝑥 ∈ On ∧ (𝐹𝑥) = 𝑦))
152, 14sylbi 207 . . 3 (𝑦 ∈ ran 𝐹 → ∃𝑥(𝑥 ∈ On ∧ (𝐹𝑥) = 𝑦))
16 nfra1 2938 . . . 4 𝑥𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))
17 nfv 1841 . . . 4 𝑥 𝑦𝐴
18 rsp 2926 . . . . 5 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → (𝑥 ∈ On → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
19 eldifi 3724 . . . . . . . 8 ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → (𝐹𝑥) ∈ 𝐴)
20 eleq1 2687 . . . . . . . 8 ((𝐹𝑥) = 𝑦 → ((𝐹𝑥) ∈ 𝐴𝑦𝐴))
2119, 20syl5ibcom 235 . . . . . . 7 ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ((𝐹𝑥) = 𝑦𝑦𝐴))
2221imim2i 16 . . . . . 6 ((𝑥 ∈ On → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) → (𝑥 ∈ On → ((𝐹𝑥) = 𝑦𝑦𝐴)))
2322impd 447 . . . . 5 ((𝑥 ∈ On → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) → ((𝑥 ∈ On ∧ (𝐹𝑥) = 𝑦) → 𝑦𝐴))
2418, 23syl 17 . . . 4 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ((𝑥 ∈ On ∧ (𝐹𝑥) = 𝑦) → 𝑦𝐴))
2516, 17, 24exlimd 2085 . . 3 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → (∃𝑥(𝑥 ∈ On ∧ (𝐹𝑥) = 𝑦) → 𝑦𝐴))
2615, 25syl5 34 . 2 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → (𝑦 ∈ ran 𝐹𝑦𝐴))
2726ssrdv 3601 1 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ran 𝐹𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wex 1702  wcel 1988  wral 2909  cdif 3564  wss 3567  cop 4174  dom cdm 5104  ran crn 5105  cima 5107  Oncon0 5711   Fn wfn 5871  cfv 5876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fn 5879  df-fv 5884
This theorem is referenced by:  tz7.48-3  7524
  Copyright terms: Public domain W3C validator