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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 8842 | . . 3 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) | |
2 | 1 | simp3bi 1148 | . 2 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) |
3 | fveq2 6843 | . . . 4 ⊢ (𝑥 = 𝐶 → (𝐹‘𝑥) = (𝐹‘𝐶)) | |
4 | fvixp.1 | . . . 4 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷) | |
5 | 3, 4 | eleq12d 2828 | . . 3 ⊢ (𝑥 = 𝐶 → ((𝐹‘𝑥) ∈ 𝐵 ↔ (𝐹‘𝐶) ∈ 𝐷)) |
6 | 5 | rspccva 3579 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐷) |
7 | 2, 6 | sylan 581 | 1 ⊢ ((𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3061 Vcvv 3444 Fn wfn 6492 ‘cfv 6497 Xcixp 8838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fn 6500 df-fv 6505 df-ixp 8839 |
This theorem is referenced by: funcf2 17759 funcpropd 17792 natcl 17845 natpropd 17870 finixpnum 36109 hspdifhsp 44943 |
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