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| Mirrors > Home > MPE Home > Th. List > fnprg | Structured version Visualization version GIF version | ||
| Description: Function with a domain of two different values. (Contributed by FL, 26-Jun-2011.) (Revised by Mario Carneiro, 26-Apr-2015.) | 
| Ref | Expression | 
|---|---|
| fnprg | ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} Fn {𝐴, 𝐵}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | funprg 6619 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}) | |
| 2 | dmpropg 6234 | . . 3 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) → dom {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐴, 𝐵}) | |
| 3 | 2 | 3ad2ant2 1134 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → dom {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐴, 𝐵}) | 
| 4 | df-fn 6563 | . 2 ⊢ ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} Fn {𝐴, 𝐵} ↔ (Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ∧ dom {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐴, 𝐵})) | |
| 5 | 1, 3, 4 | sylanbrc 583 | 1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} Fn {𝐴, 𝐵}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 {cpr 4627 〈cop 4631 dom cdm 5684 Fun wfun 6554 Fn wfn 6555 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-fun 6562 df-fn 6563 | 
| This theorem is referenced by: f1oprswap 6891 fnpr2o 17603 zlmodzxzscm 48278 zlmodzxzadd 48279 | 
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