Theorem fvmptdv2 5766

Description: Alternate deduction version of fvmpt 5753, 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 2233	. . 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 2824	. . . . . 6 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ V)
5	3, 4	syl 14	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
6		isset 2819	. . . . 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 2824	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V)
10	8, 9	syl 14	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ V)
11	2, 10	eqeltrrd 2310	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 ∈ V)
12	7, 11	exlimddv 1948	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
13	1, 2, 3, 12	fvmptd 5757	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶)
14		fveq1 5668	. . 3 ⊢ (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴))
15	14	eqeq1d 2241	. 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 1398  ∃wex 1541   ∈ wcel 2203  Vcvv 2812   ↦ cmpt 4170  ‘cfv 5351

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-sbc 3042  df-csb 3138  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-iota 5311  df-fun 5353  df-fv 5359

This theorem is referenced by: (None)