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Mirrors > Home > MPE Home > Th. List > xpsc | Structured version Visualization version GIF version |
Description: A short expression for the pair function mapping 0 to 𝐴 and 1 to 𝐵. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
xpsc | ⊢ ◡({𝐴} +𝑐 {𝐵}) = (({∅} × {𝐴}) ∪ ({1𝑜} × {𝐵})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex 4908 | . . . 4 ⊢ {𝐴} ∈ V | |
2 | snex 4908 | . . . 4 ⊢ {𝐵} ∈ V | |
3 | cdaval 8992 | . . . 4 ⊢ (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} +𝑐 {𝐵}) = (({𝐴} × {∅}) ∪ ({𝐵} × {1𝑜}))) | |
4 | 1, 2, 3 | mp2an 708 | . . 3 ⊢ ({𝐴} +𝑐 {𝐵}) = (({𝐴} × {∅}) ∪ ({𝐵} × {1𝑜})) |
5 | 4 | cnveqi 5297 | . 2 ⊢ ◡({𝐴} +𝑐 {𝐵}) = ◡(({𝐴} × {∅}) ∪ ({𝐵} × {1𝑜})) |
6 | cnvun 5538 | . 2 ⊢ ◡(({𝐴} × {∅}) ∪ ({𝐵} × {1𝑜})) = (◡({𝐴} × {∅}) ∪ ◡({𝐵} × {1𝑜})) | |
7 | cnvxp 5551 | . . 3 ⊢ ◡({𝐴} × {∅}) = ({∅} × {𝐴}) | |
8 | cnvxp 5551 | . . 3 ⊢ ◡({𝐵} × {1𝑜}) = ({1𝑜} × {𝐵}) | |
9 | 7, 8 | uneq12i 3765 | . 2 ⊢ (◡({𝐴} × {∅}) ∪ ◡({𝐵} × {1𝑜})) = (({∅} × {𝐴}) ∪ ({1𝑜} × {𝐵})) |
10 | 5, 6, 9 | 3eqtri 2648 | 1 ⊢ ◡({𝐴} +𝑐 {𝐵}) = (({∅} × {𝐴}) ∪ ({1𝑜} × {𝐵})) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∪ cun 3572 ∅c0 3915 {csn 4177 × cxp 5112 ◡ccnv 5113 (class class class)co 6650 1𝑜c1o 7553 +𝑐 ccda 8989 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-cda 8990 |
This theorem is referenced by: xpscg 16218 xpsc0 16220 xpsc1 16221 xpsfrnel2 16225 |
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