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Mirrors > Home > MPE Home > Th. List > funprg | Structured version Visualization version GIF version |
Description: A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.) (Proof shortened by JJ, 14-Jul-2021.) |
Ref | Expression |
---|---|
funprg | ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funsng 6405 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) → Fun {〈𝐴, 𝐶〉}) | |
2 | funsng 6405 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐷 ∈ 𝑌) → Fun {〈𝐵, 𝐷〉}) | |
3 | 1, 2 | anim12i 614 | . . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑋) ∧ (𝐵 ∈ 𝑊 ∧ 𝐷 ∈ 𝑌)) → (Fun {〈𝐴, 𝐶〉} ∧ Fun {〈𝐵, 𝐷〉})) |
4 | 3 | an4s 658 | . . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → (Fun {〈𝐴, 𝐶〉} ∧ Fun {〈𝐵, 𝐷〉})) |
5 | 4 | 3adant3 1128 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → (Fun {〈𝐴, 𝐶〉} ∧ Fun {〈𝐵, 𝐷〉})) |
6 | dmsnopg 6070 | . . . . . 6 ⊢ (𝐶 ∈ 𝑋 → dom {〈𝐴, 𝐶〉} = {𝐴}) | |
7 | dmsnopg 6070 | . . . . . 6 ⊢ (𝐷 ∈ 𝑌 → dom {〈𝐵, 𝐷〉} = {𝐵}) | |
8 | 6, 7 | ineqan12d 4191 | . . . . 5 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) → (dom {〈𝐴, 𝐶〉} ∩ dom {〈𝐵, 𝐷〉}) = ({𝐴} ∩ {𝐵})) |
9 | disjsn2 4648 | . . . . 5 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | |
10 | 8, 9 | sylan9eq 2876 | . . . 4 ⊢ (((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → (dom {〈𝐴, 𝐶〉} ∩ dom {〈𝐵, 𝐷〉}) = ∅) |
11 | 10 | 3adant1 1126 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → (dom {〈𝐴, 𝐶〉} ∩ dom {〈𝐵, 𝐷〉}) = ∅) |
12 | funun 6400 | . . 3 ⊢ (((Fun {〈𝐴, 𝐶〉} ∧ Fun {〈𝐵, 𝐷〉}) ∧ (dom {〈𝐴, 𝐶〉} ∩ dom {〈𝐵, 𝐷〉}) = ∅) → Fun ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉})) | |
13 | 5, 11, 12 | syl2anc 586 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → Fun ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉})) |
14 | df-pr 4570 | . . 3 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉}) | |
15 | 14 | funeqi 6376 | . 2 ⊢ (Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ↔ Fun ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐷〉})) |
16 | 13, 15 | sylibr 236 | 1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∪ cun 3934 ∩ cin 3935 ∅c0 4291 {csn 4567 {cpr 4569 〈cop 4573 dom cdm 5555 Fun wfun 6349 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-fun 6357 |
This theorem is referenced by: funtpg 6409 funpr 6410 fnprg 6413 fpropnf1 7025 |
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