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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 8467 | . . 3 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) | |
2 | 1 | simp3bi 1143 | . 2 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) |
3 | fveq2 6672 | . . . 4 ⊢ (𝑥 = 𝐶 → (𝐹‘𝑥) = (𝐹‘𝐶)) | |
4 | fvixp.1 | . . . 4 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷) | |
5 | 3, 4 | eleq12d 2909 | . . 3 ⊢ (𝑥 = 𝐶 → ((𝐹‘𝑥) ∈ 𝐵 ↔ (𝐹‘𝐶) ∈ 𝐷)) |
6 | 5 | rspccva 3624 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐷) |
7 | 2, 6 | sylan 582 | 1 ⊢ ((𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 Fn wfn 6352 ‘cfv 6357 Xcixp 8463 |
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 2795 |
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-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fn 6360 df-fv 6365 df-ixp 8464 |
This theorem is referenced by: funcf2 17140 funcpropd 17172 natcl 17225 natpropd 17248 finixpnum 34879 hspdifhsp 42905 |
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