Theorem tfrlem3-2d 6573

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 5690	. . . . . 6 ⊢ (𝑥 = 𝑔 → (𝐹‘𝑥) = (𝐹‘𝑔))
3	2	eleq1d 2307	. . . . 5 ⊢ (𝑥 = 𝑔 → ((𝐹‘𝑥) ∈ V ↔ (𝐹‘𝑔) ∈ V))
4	3	anbi2d 468	. . . 4 ⊢ (𝑥 = 𝑔 → ((Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V) ↔ (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V)))
5	4	cbvalv 1973	. . 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 1611	1 ⊢ (𝜑 → (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1400   ∈ wcel 2209  Vcvv 2821  Fun wfun 5366  ‘cfv 5372

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-iota 5332  df-fv 5380

This theorem is referenced by:  tfrlemisucfn  6585  tfrlemisucaccv  6586  tfrlemibxssdm  6588  tfrlemibfn  6589  tfrlemi14d  6594