Theorem fvmptd3f 6872

Description: Alternate deduction version of fvmpt 6857 with three nonfreeness hypotheses instead of distinct variable conditions. (Contributed 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))
fvmptd3f.4	⊢ Ⅎ𝑥𝐹
fvmptd3f.5	⊢ Ⅎ𝑥𝜓
fvmptd3f.6	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
fvmptd3f	⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐷

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

Proof of Theorem fvmptd3f

Step	Hyp	Ref	Expression
1		fvmptd3f.6	. 2 ⊢ Ⅎ𝑥𝜑
2		fvmptd3f.4	. . . 4 ⊢ Ⅎ𝑥𝐹
3		nfmpt1 5178	. . . 4 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐷 ↦ 𝐵)
4	2, 3	nfeq 2919	. . 3 ⊢ Ⅎ𝑥 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)
5		fvmptd3f.5	. . 3 ⊢ Ⅎ𝑥𝜓
6	4, 5	nfim 1900	. 2 ⊢ Ⅎ𝑥(𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓)
7		fvmptd2f.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐷)
8	7	elexd 3442	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
9		isset 3435	. . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
10	8, 9	sylib 217	. 2 ⊢ (𝜑 → ∃𝑥 𝑥 = 𝐴)
11		fveq1 6755	. . 3 ⊢ (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴))
12		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴)
13	12	fveq2d 6760	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴))
14	7	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐴 ∈ 𝐷)
15	12, 14	eqeltrd 2839	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 ∈ 𝐷)
16		fvmptd2f.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉)
17		eqid 2738	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 ↦ 𝐵) = (𝑥 ∈ 𝐷 ↦ 𝐵)
18	17	fvmpt2 6868	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝑥) = 𝐵)
19	15, 16, 18	syl2anc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝑥) = 𝐵)
20	13, 19	eqtr3d 2780	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐵)
21	20	eqeq2d 2749	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) ↔ (𝐹‘𝐴) = 𝐵))
22		fvmptd2f.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = 𝐵 → 𝜓))
23	21, 22	sylbid 239	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) → 𝜓))
24	11, 23	syl5 34	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓))
25	1, 6, 10, 24	exlimdd 2216	1 ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539  ∃wex 1783  Ⅎwnf 1787   ∈ wcel 2108  Ⅎwnfc 2886  Vcvv 3422   ↦ cmpt 5153  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fv 6426

This theorem is referenced by:  fvmptd2f  6873