Theorem fvmptdv2 7004

Description: Alternate deduction version of fvmpt 6986, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Hypotheses

Ref	Expression
fvmptdv2.1	⊢ (𝜑 → 𝐴 ∈ 𝐷)
fvmptdv2.2	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉)
fvmptdv2.3	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)

Assertion

Ref	Expression
fvmptdv2	⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → (𝐹‘𝐴) = 𝐶))

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

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

Proof of Theorem fvmptdv2

Step	Hyp	Ref	Expression
1		eqidd 2736	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ 𝐵) = (𝑥 ∈ 𝐷 ↦ 𝐵))
2		fvmptdv2.3	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)
3		fvmptdv2.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐷)
4	3	elexd 3483	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
5		isset 3473	. . . . 5 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
6	4, 5	sylib 218	. . . 4 ⊢ (𝜑 → ∃𝑥 𝑥 = 𝐴)
7		fvmptdv2.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉)
8	7	elexd 3483	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ V)
9	2, 8	eqeltrrd 2835	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 ∈ V)
10	6, 9	exlimddv 1935	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
11	1, 2, 3, 10	fvmptd 6993	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶)
12		fveq1 6875	. . 3 ⊢ (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴))
13	12	eqeq1d 2737	. 2 ⊢ (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → ((𝐹‘𝐴) = 𝐶 ↔ ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶))
14	11, 13	syl5ibrcom 247	1 ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → (𝐹‘𝐴) = 𝐶))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  Vcvv 3459   ↦ cmpt 5201  ‘cfv 6531

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-iota 6484  df-fun 6533  df-fv 6539

This theorem is referenced by:  curf12  18239  curf2  18241  yonedalem4b  18288