Theorem tfrlem3-2d 6315

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 5517	. . . . . 6 ⊢ (𝑥 = 𝑔 → (𝐹‘𝑥) = (𝐹‘𝑔))
3	2	eleq1d 2246	. . . . 5 ⊢ (𝑥 = 𝑔 → ((𝐹‘𝑥) ∈ V ↔ (𝐹‘𝑔) ∈ V))
4	3	anbi2d 464	. . . 4 ⊢ (𝑥 = 𝑔 → ((Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V) ↔ (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V)))
5	4	cbvalv 1917	. . 3 ⊢ (∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V) ↔ ∀𝑔(Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V))
6	1, 5	sylib 122	. 2 ⊢ (𝜑 → ∀𝑔(Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V))
7	6	19.21bi 1558	1 ⊢ (𝜑 → (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1351   ∈ wcel 2148  Vcvv 2739  Fun wfun 5212  ‘cfv 5218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226

This theorem is referenced by:  tfrlemisucfn  6327  tfrlemisucaccv  6328  tfrlemibxssdm  6330  tfrlemibfn  6331  tfrlemi14d  6336