Theorem fvmptd2f 6950

Description: Alternate deduction version of fvmpt 6934, 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 1914	. 2 ⊢ Ⅎ𝑥𝜑
7	1, 2, 3, 4, 5, 6	fvmptd3f 6949	1 ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2109  Ⅎwnfc 2876   ↦ cmpt 5176  ‘cfv 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fv 6494

This theorem is referenced by:  fvmptdv  6951  yonedalem4b  18200