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Mirrors > Home > ILE Home > Th. List > fvmptdv2 | GIF version |
Description: Alternate deduction version of fvmpt 5592, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.) |
Ref | Expression |
---|---|
fvmptdv2.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐷) |
fvmptdv2.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉) |
fvmptdv2.3 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
fvmptdv2 | ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → (𝐹‘𝐴) = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2178 | . . 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 2748 | . . . . . 6 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ V) | |
5 | 3, 4 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ V) |
6 | isset 2743 | . . . . 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 2748 | . . . . . 6 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V) | |
10 | 8, 9 | syl 14 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ V) |
11 | 2, 10 | eqeltrrd 2255 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 ∈ V) |
12 | 7, 11 | exlimddv 1898 | . . 3 ⊢ (𝜑 → 𝐶 ∈ V) |
13 | 1, 2, 3, 12 | fvmptd 5596 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶) |
14 | fveq1 5513 | . . 3 ⊢ (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴)) | |
15 | 14 | eqeq1d 2186 | . 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)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∃wex 1492 ∈ wcel 2148 Vcvv 2737 ↦ cmpt 4063 ‘cfv 5215 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-iota 5177 df-fun 5217 df-fv 5223 |
This theorem is referenced by: (None) |
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