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Theorem tz7.44-1 8407
Description: The value of 𝐹 at . Part 1 of Theorem 7.44 of [TakeutiZaring] p. 49. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
Hypotheses
Ref Expression
tz7.44.1 𝐺 = (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ran 𝑥, (𝐻‘(𝑥 dom 𝑥)))))
tz7.44.2 (𝑦𝑋 → (𝐹𝑦) = (𝐺‘(𝐹𝑦)))
tz7.44-1.3 𝐴 ∈ V
Assertion
Ref Expression
tz7.44-1 (∅ ∈ 𝑋 → (𝐹‘∅) = 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐹   𝑦,𝐺   𝑥,𝐻   𝑦,𝑋
Allowed substitution hints:   𝐴(𝑦)   𝐺(𝑥)   𝐻(𝑦)   𝑋(𝑥)

Proof of Theorem tz7.44-1
StepHypRef Expression
1 fveq2 6885 . . . 4 (𝑦 = ∅ → (𝐹𝑦) = (𝐹‘∅))
2 reseq2 5970 . . . . . 6 (𝑦 = ∅ → (𝐹𝑦) = (𝐹 ↾ ∅))
3 res0 5979 . . . . . 6 (𝐹 ↾ ∅) = ∅
42, 3eqtrdi 2782 . . . . 5 (𝑦 = ∅ → (𝐹𝑦) = ∅)
54fveq2d 6889 . . . 4 (𝑦 = ∅ → (𝐺‘(𝐹𝑦)) = (𝐺‘∅))
61, 5eqeq12d 2742 . . 3 (𝑦 = ∅ → ((𝐹𝑦) = (𝐺‘(𝐹𝑦)) ↔ (𝐹‘∅) = (𝐺‘∅)))
7 tz7.44.2 . . 3 (𝑦𝑋 → (𝐹𝑦) = (𝐺‘(𝐹𝑦)))
86, 7vtoclga 3560 . 2 (∅ ∈ 𝑋 → (𝐹‘∅) = (𝐺‘∅))
9 0ex 5300 . . 3 ∅ ∈ V
10 iftrue 4529 . . . 4 (𝑥 = ∅ → if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ran 𝑥, (𝐻‘(𝑥 dom 𝑥)))) = 𝐴)
11 tz7.44.1 . . . 4 𝐺 = (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ran 𝑥, (𝐻‘(𝑥 dom 𝑥)))))
12 tz7.44-1.3 . . . 4 𝐴 ∈ V
1310, 11, 12fvmpt 6992 . . 3 (∅ ∈ V → (𝐺‘∅) = 𝐴)
149, 13ax-mp 5 . 2 (𝐺‘∅) = 𝐴
158, 14eqtrdi 2782 1 (∅ ∈ 𝑋 → (𝐹‘∅) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3468  c0 4317  ifcif 4523   cuni 4902  cmpt 5224  dom cdm 5669  ran crn 5670  cres 5671  Lim wlim 6359  cfv 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-res 5681  df-iota 6489  df-fun 6539  df-fv 6545
This theorem is referenced by:  rdg0  8422
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