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Theorem tz7.44-1 8435
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 6902 . . . 4 (𝑦 = ∅ → (𝐹𝑦) = (𝐹‘∅))
2 reseq2 5984 . . . . . 6 (𝑦 = ∅ → (𝐹𝑦) = (𝐹 ↾ ∅))
3 res0 5993 . . . . . 6 (𝐹 ↾ ∅) = ∅
42, 3eqtrdi 2784 . . . . 5 (𝑦 = ∅ → (𝐹𝑦) = ∅)
54fveq2d 6906 . . . 4 (𝑦 = ∅ → (𝐺‘(𝐹𝑦)) = (𝐺‘∅))
61, 5eqeq12d 2744 . . 3 (𝑦 = ∅ → ((𝐹𝑦) = (𝐺‘(𝐹𝑦)) ↔ (𝐹‘∅) = (𝐺‘∅)))
7 tz7.44.2 . . 3 (𝑦𝑋 → (𝐹𝑦) = (𝐺‘(𝐹𝑦)))
86, 7vtoclga 3565 . 2 (∅ ∈ 𝑋 → (𝐹‘∅) = (𝐺‘∅))
9 0ex 5311 . . 3 ∅ ∈ V
10 iftrue 4538 . . . 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 7010 . . 3 (∅ ∈ V → (𝐺‘∅) = 𝐴)
149, 13ax-mp 5 . 2 (𝐺‘∅) = 𝐴
158, 14eqtrdi 2784 1 (∅ ∈ 𝑋 → (𝐹‘∅) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3473  c0 4326  ifcif 4532   cuni 4912  cmpt 5235  dom cdm 5682  ran crn 5683  cres 5684  Lim wlim 6375  cfv 6553
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-res 5694  df-iota 6505  df-fun 6555  df-fv 6561
This theorem is referenced by:  rdg0  8450
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