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Mirrors > Home > MPE Home > Th. List > fvixp | Structured version Visualization version GIF version |
Description: Projection of a factor of an indexed Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.) |
Ref | Expression |
---|---|
fvixp.1 | ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
fvixp | ⊢ ((𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elixp2 8892 | . . 3 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) | |
2 | 1 | simp3bi 1144 | . 2 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) |
3 | fveq2 6882 | . . . 4 ⊢ (𝑥 = 𝐶 → (𝐹‘𝑥) = (𝐹‘𝐶)) | |
4 | fvixp.1 | . . . 4 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷) | |
5 | 3, 4 | eleq12d 2819 | . . 3 ⊢ (𝑥 = 𝐶 → ((𝐹‘𝑥) ∈ 𝐵 ↔ (𝐹‘𝐶) ∈ 𝐷)) |
6 | 5 | rspccva 3603 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐷) |
7 | 2, 6 | sylan 579 | 1 ⊢ ((𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3053 Vcvv 3466 Fn wfn 6529 ‘cfv 6534 Xcixp 8888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-iota 6486 df-fun 6536 df-fn 6537 df-fv 6542 df-ixp 8889 |
This theorem is referenced by: funcf2 17823 funcpropd 17858 natcl 17912 natpropd 17937 finixpnum 36976 hspdifhsp 45877 |
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