Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elixpconstg | Structured version Visualization version GIF version |
Description: Membership in an infinite Cartesian product of a constant 𝐵. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
Ref | Expression |
---|---|
elixpconstg | ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ 𝐹:𝐴⟶𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ixpfn 8469 | . . 3 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐹 Fn 𝐴) | |
2 | elixp2 8467 | . . . 4 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) | |
3 | 2 | simp3bi 1143 | . . 3 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) |
4 | ffnfv 6884 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) | |
5 | 1, 3, 4 | sylanbrc 585 | . 2 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐹:𝐴⟶𝐵) |
6 | elex 3514 | . . . . 5 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V) | |
7 | 6 | adantr 483 | . . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶𝐵) → 𝐹 ∈ V) |
8 | ffn 6516 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
9 | 8 | adantl 484 | . . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶𝐵) → 𝐹 Fn 𝐴) |
10 | 4 | simprbi 499 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) |
11 | 10 | adantl 484 | . . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶𝐵) → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) |
12 | 7, 9, 11, 2 | syl3anbrc 1339 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴⟶𝐵) → 𝐹 ∈ X𝑥 ∈ 𝐴 𝐵) |
13 | 12 | ex 415 | . 2 ⊢ (𝐹 ∈ 𝑉 → (𝐹:𝐴⟶𝐵 → 𝐹 ∈ X𝑥 ∈ 𝐴 𝐵)) |
14 | 5, 13 | impbid2 228 | 1 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ 𝐹:𝐴⟶𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 Fn wfn 6352 ⟶wf 6353 ‘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 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 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-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ixp 8464 |
This theorem is referenced by: iinhoiicclem 42962 |
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