Theorem tfrlem3-2d 6280

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 5486	. . . . . 6 ⊢ (𝑥 = 𝑔 → (𝐹‘𝑥) = (𝐹‘𝑔))
3	2	eleq1d 2235	. . . . 5 ⊢ (𝑥 = 𝑔 → ((𝐹‘𝑥) ∈ V ↔ (𝐹‘𝑔) ∈ V))
4	3	anbi2d 460	. . . 4 ⊢ (𝑥 = 𝑔 → ((Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V) ↔ (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V)))
5	4	cbvalv 1905	. . 3 ⊢ (∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V) ↔ ∀𝑔(Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V))
6	1, 5	sylib 121	. 2 ⊢ (𝜑 → ∀𝑔(Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V))
7	6	19.21bi 1546	1 ⊢ (𝜑 → (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V))

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1341   ∈ wcel 2136  Vcvv 2726  Fun wfun 5182  ‘cfv 5188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-iota 5153  df-fv 5196

This theorem is referenced by:  tfrlemisucfn  6292  tfrlemisucaccv  6293  tfrlemibxssdm  6295  tfrlemibfn  6296  tfrlemi14d  6301