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Mirrors > Home > MPE Home > Th. List > fvmptdv2 | Structured version Visualization version GIF version |
Description: Alternate deduction version of fvmpt 6768, 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 2822 | . . 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 3514 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ V) |
5 | isset 3506 | . . . . 5 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
6 | 4, 5 | sylib 220 | . . . 4 ⊢ (𝜑 → ∃𝑥 𝑥 = 𝐴) |
7 | fvmptdv2.2 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉) | |
8 | 7 | elexd 3514 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ V) |
9 | 2, 8 | eqeltrrd 2914 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 ∈ V) |
10 | 6, 9 | exlimddv 1936 | . . 3 ⊢ (𝜑 → 𝐶 ∈ V) |
11 | 1, 2, 3, 10 | fvmptd 6775 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶) |
12 | fveq1 6669 | . . 3 ⊢ (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴)) | |
13 | 12 | eqeq1d 2823 | . 2 ⊢ (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → ((𝐹‘𝐴) = 𝐶 ↔ ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶)) |
14 | 11, 13 | syl5ibrcom 249 | 1 ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → (𝐹‘𝐴) = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 Vcvv 3494 ↦ cmpt 5146 ‘cfv 6355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 |
This theorem is referenced by: curf12 17477 curf2 17479 yonedalem4b 17526 |
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