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| Mirrors > Home > MPE Home > Th. List > xpima2 | Structured version Visualization version GIF version | ||
| Description: Direct image by a Cartesian product (case of nonempty intersection with the domain). (Contributed by Thierry Arnoux, 16-Dec-2017.) |
| Ref | Expression |
|---|---|
| xpima2 | ⊢ ((𝐴 ∩ 𝐶) ≠ ∅ → ((𝐴 × 𝐵) “ 𝐶) = 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | xpima 6202 | . 2 ⊢ ((𝐴 × 𝐵) “ 𝐶) = if((𝐴 ∩ 𝐶) = ∅, ∅, 𝐵) | |
| 2 | ifnefalse 4537 | . 2 ⊢ ((𝐴 ∩ 𝐶) ≠ ∅ → if((𝐴 ∩ 𝐶) = ∅, ∅, 𝐵) = 𝐵) | |
| 3 | 1, 2 | eqtrid 2789 | 1 ⊢ ((𝐴 ∩ 𝐶) ≠ ∅ → ((𝐴 × 𝐵) “ 𝐶) = 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ≠ wne 2940 ∩ cin 3950 ∅c0 4333 ifcif 4525 × cxp 5683 “ cima 5688 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 |
| This theorem is referenced by: xpimasn 6205 ustuqtop1 24250 ustuqtop5 24254 prjcrv0 42643 brtrclfv2 43740 aacllem 49320 |
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