Theorem fvmptdv2 5651

Description: Alternate deduction version of fvmpt 5638, 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 2197	. . 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		elex 2774	. . . . . 6 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ V)
5	3, 4	syl 14	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
6		isset 2769	. . . . 5 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
7	5, 6	sylib 122	. . . 4 ⊢ (𝜑 → ∃𝑥 𝑥 = 𝐴)
8		fvmptdv2.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉)
9		elex 2774	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V)
10	8, 9	syl 14	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ V)
11	2, 10	eqeltrrd 2274	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 ∈ V)
12	7, 11	exlimddv 1913	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
13	1, 2, 3, 12	fvmptd 5642	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶)
14		fveq1 5557	. . 3 ⊢ (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴))
15	14	eqeq1d 2205	. 2 ⊢ (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → ((𝐹‘𝐴) = 𝐶 ↔ ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶))
16	13, 15	syl5ibrcom 157	1 ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → (𝐹‘𝐴) = 𝐶))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364  ∃wex 1506   ∈ wcel 2167  Vcvv 2763   ↦ cmpt 4094  ‘cfv 5258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266

This theorem is referenced by: (None)