Theorem tfrlem3-2 5927

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 5178	. . . 4 ⊢ (𝑥 = 𝑔 → (𝐹‘𝑥) = (𝐹‘𝑔))
2	1	eleq1d 2106	. . 3 ⊢ (𝑥 = 𝑔 → ((𝐹‘𝑥) ∈ V ↔ (𝐹‘𝑔) ∈ V))
3	2	anbi2d 437	. 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 1812	1 ⊢ (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V)

Syntax hints:   ∧ wa 97   = wceq 1243   ∈ wcel 1393  Vcvv 2557  Fun wfun 4896  ‘cfv 4902

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910

This theorem is referenced by: (None)