Theorem tfrlem3-2d 5869

Description: Lemma for transfinite recursion which changes a bound variable (Contributed by Jim Kingdon, 2-Jul-2019.)

Hypothesis

Ref	Expression
tfrlem3-2d.1	⊢ (φ → ∀x(Fun 𝐹 ∧ (𝐹‘x) ∈ V))

Assertion

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

Distinct variable group:   x,g,𝐹

Allowed substitution hints:   φ(x,g)

Proof of Theorem tfrlem3-2d

Step	Hyp	Ref	Expression
1		tfrlem3-2d.1	. . 3 ⊢ (φ → ∀x(Fun 𝐹 ∧ (𝐹‘x) ∈ V))
2		fveq2 5121	. . . . . 6 ⊢ (x = g → (𝐹‘x) = (𝐹‘g))
3	2	eleq1d 2103	. . . . 5 ⊢ (x = g → ((𝐹‘x) ∈ V ↔ (𝐹‘g) ∈ V))
4	3	anbi2d 437	. . . 4 ⊢ (x = g → ((Fun 𝐹 ∧ (𝐹‘x) ∈ V) ↔ (Fun 𝐹 ∧ (𝐹‘g) ∈ V)))
5	4	cbvalv 1791	. . 3 ⊢ (∀x(Fun 𝐹 ∧ (𝐹‘x) ∈ V) ↔ ∀g(Fun 𝐹 ∧ (𝐹‘g) ∈ V))
6	1, 5	sylib 127	. 2 ⊢ (φ → ∀g(Fun 𝐹 ∧ (𝐹‘g) ∈ V))
7	6	19.21bi 1447	1 ⊢ (φ → (Fun 𝐹 ∧ (𝐹‘g) ∈ V))

Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1240   ∈ wcel 1390  Vcvv 2551  Fun wfun 4839  ‘cfv 4845

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-iota 4810  df-fv 4853

This theorem is referenced by:  tfrlemisucfn  5879  tfrlemisucaccv  5880  tfrlemibxssdm  5882  tfrlemibfn  5883  tfrlemi14d  5888