Theorem tfrlem3-2 5950

Description: Lemma for transfinite recursion which changes a bound variable (Contributed by Jim Kingdon, 17-Apr-2019.)

Hypothesis

Ref	Expression
tfrlem3-2.1	⊢ (Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V)

Assertion

Ref	Expression
tfrlem3-2	⊢ (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V)

Distinct variable group:   𝑥,𝑔,𝐹

Proof of Theorem tfrlem3-2

Step	Hyp	Ref	Expression
1		fveq2 5198	. . . 4 ⊢ (𝑥 = 𝑔 → (𝐹‘𝑥) = (𝐹‘𝑔))
2	1	eleq1d 2147	. . 3 ⊢ (𝑥 = 𝑔 → ((𝐹‘𝑥) ∈ V ↔ (𝐹‘𝑔) ∈ V))
3	2	anbi2d 451	. 2 ⊢ (𝑥 = 𝑔 → ((Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V) ↔ (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V)))
4		tfrlem3-2.1	. 2 ⊢ (Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V)
5	3, 4	chvarv 1853	1 ⊢ (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V)

Syntax hints:   ∧ wa 102   = wceq 1284   ∈ wcel 1433  Vcvv 2601  Fun wfun 4916  ‘cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-iota 4887  df-fv 4930

This theorem is referenced by: (None)