Theorem fvmptd2f 7007

Description: Alternate deduction version of fvmpt 6991, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.) (Proof shortened by AV, 19-Jan-2022.)

Hypotheses

Ref	Expression
fvmptd2f.1	⊢ (𝜑 → 𝐴 ∈ 𝐷)
fvmptd2f.2	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉)
fvmptd2f.3	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = 𝐵 → 𝜓))
fvmptd2f.4	⊢ Ⅎ𝑥𝐹
fvmptd2f.5	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
fvmptd2f	⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐷   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem fvmptd2f

Step	Hyp	Ref	Expression
1		fvmptd2f.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐷)
2		fvmptd2f.2	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉)
3		fvmptd2f.3	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = 𝐵 → 𝜓))
4		fvmptd2f.4	. 2 ⊢ Ⅎ𝑥𝐹
5		fvmptd2f.5	. 2 ⊢ Ⅎ𝑥𝜓
6		nfv 1909	. 2 ⊢ Ⅎ𝑥𝜑
7	1, 2, 3, 4, 5, 6	fvmptd3f 7006	1 ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533  Ⅎwnf 1777   ∈ wcel 2098  Ⅎwnfc 2877   ↦ cmpt 5224  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fv 6544

This theorem is referenced by:  fvmptdv  7008  yonedalem4b  18238