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Mirrors > Home > MPE Home > Th. List > fvmptdv2 | Structured version Visualization version GIF version |
Description: Alternate deduction version of fvmpt 6191, 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 2611 | . . 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 3185 | . . . . . 6 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ V) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ V) |
6 | isset 3180 | . . . . 5 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
7 | 5, 6 | sylib 207 | . . . 4 ⊢ (𝜑 → ∃𝑥 𝑥 = 𝐴) |
8 | fvmptdv2.2 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉) | |
9 | elex 3185 | . . . . . 6 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V) | |
10 | 8, 9 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ V) |
11 | 2, 10 | eqeltrrd 2689 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 ∈ V) |
12 | 7, 11 | exlimddv 1850 | . . 3 ⊢ (𝜑 → 𝐶 ∈ V) |
13 | 1, 2, 3, 12 | fvmptd 6197 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶) |
14 | fveq1 6102 | . . 3 ⊢ (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴)) | |
15 | 14 | eqeq1d 2612 | . 2 ⊢ (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → ((𝐹‘𝐴) = 𝐶 ↔ ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐶)) |
16 | 13, 15 | syl5ibrcom 236 | 1 ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → (𝐹‘𝐴) = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 Vcvv 3173 ↦ cmpt 4643 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 |
This theorem is referenced by: curf12 16690 curf2 16692 yonedalem4b 16739 |
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