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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 6239 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → Fun {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉}) | |
2 | dmpropg 5909 | . . 3 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) → dom {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐴, 𝐵}) | |
3 | 2 | 3ad2ant2 1115 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → dom {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐴, 𝐵}) |
4 | df-fn 6189 | . 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 575 | 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 387 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ≠ wne 2962 {cpr 4438 〈cop 4442 dom cdm 5404 Fun wfun 6180 Fn wfn 6181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pr 5183 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-ral 3088 df-rex 3089 df-rab 3092 df-v 3412 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-sn 4437 df-pr 4439 df-op 4443 df-br 4927 df-opab 4989 df-id 5309 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-fun 6188 df-fn 6189 |
This theorem is referenced by: f1oprswap 6485 xpscfn 16687 fnpr2o 16688 zlmodzxzscm 43799 zlmodzxzadd 43800 |
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