Theorem fvmptd3f 7012

Description: Alternate deduction version of fvmpt 6996 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 5215	. . . 4 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐷 ↦ 𝐵)
4	2, 3	nfeq 2941	. . 3 ⊢ Ⅎ𝑥 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)
5		fvmptd3f.5	. . 3 ⊢ Ⅎ𝑥𝜓
6	4, 5	nfim 1929	. 2 ⊢ Ⅎ𝑥(𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓)
7		fvmptd2f.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐷)
8	7	elexd 3481	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
9		isset 3472	. . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
10	8, 9	sylib 221	. 2 ⊢ (𝜑 → ∃𝑥 𝑥 = 𝐴)
11		fveq1 6887	. . 3 ⊢ (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴))
12		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴)
13	12	fveq2d 6892	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴))
14	7	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐴 ∈ 𝐷)
15	12, 14	eqeltrd 2866	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 ∈ 𝐷)
16		fvmptd2f.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉)
17		eqid 2766	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 ↦ 𝐵) = (𝑥 ∈ 𝐷 ↦ 𝐵)
18	17	fvmpt2 7008	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝑥) = 𝐵)
19	15, 16, 18	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝑥) = 𝐵)
20	13, 19	eqtr3d 2803	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐵)
21	20	eqeq2d 2777	. . . 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 243	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) → 𝜓))
24	11, 23	syl5 35	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓))
25	1, 6, 10, 24	exlimdd 2259	1 ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812  Ⅎwnf 1816   ∈ wcel 2146  Ⅎwnfc 2913  Vcvv 3458   ↦ cmpt 5197  ‘cfv 6543

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fv 6551

This theorem is used by:  fvmptd2f  7013