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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fprmappr | Structured version Visualization version GIF version |
Description: A function with a domain of two elements as element of the mapping operator applied to a pair. (Contributed by AV, 20-May-2024.) |
Ref | Expression |
---|---|
fprmappr | ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ∈ (𝑋 ↑m {𝐴, 𝐵})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3simpa 1147 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑊)) | |
2 | 1 | adantr 480 | . . . . 5 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑊)) |
3 | simpr 484 | . . . . 5 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) | |
4 | simpl3 1192 | . . . . 5 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → 𝐴 ≠ 𝐵) | |
5 | fprg 7175 | . . . . 5 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) ∧ 𝐴 ≠ 𝐵) → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}:{𝐴, 𝐵}⟶{𝐶, 𝐷}) | |
6 | 2, 3, 4, 5 | syl3anc 1370 | . . . 4 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}:{𝐴, 𝐵}⟶{𝐶, 𝐷}) |
7 | prssi 4826 | . . . . 5 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) → {𝐶, 𝐷} ⊆ 𝑋) | |
8 | 7 | adantl 481 | . . . 4 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → {𝐶, 𝐷} ⊆ 𝑋) |
9 | 6, 8 | fssd 6754 | . . 3 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}:{𝐴, 𝐵}⟶𝑋) |
10 | 9 | 3adant1 1129 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}:{𝐴, 𝐵}⟶𝑋) |
11 | simp1 1135 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → 𝑋 ∈ 𝑉) | |
12 | prex 5443 | . . . 4 ⊢ {𝐴, 𝐵} ∈ V | |
13 | 12 | a1i 11 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → {𝐴, 𝐵} ∈ V) |
14 | 11, 13 | elmapd 8879 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ∈ (𝑋 ↑m {𝐴, 𝐵}) ↔ {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}:{𝐴, 𝐵}⟶𝑋)) |
15 | 10, 14 | mpbird 257 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ∈ (𝑋 ↑m {𝐴, 𝐵})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ⊆ wss 3963 {cpr 4633 〈cop 4637 ⟶wf 6559 (class class class)co 7431 ↑m cmap 8865 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8867 |
This theorem is referenced by: mapprop 48191 fv2arycl 48498 2arymptfv 48500 |
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