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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 6025 | . 2 ⊢ ((𝐴 × 𝐵) “ 𝐶) = if((𝐴 ∩ 𝐶) = ∅, ∅, 𝐵) | |
2 | ifnefalse 4437 | . 2 ⊢ ((𝐴 ∩ 𝐶) ≠ ∅ → if((𝐴 ∩ 𝐶) = ∅, ∅, 𝐵) = 𝐵) | |
3 | 1, 2 | syl5eq 2783 | 1 ⊢ ((𝐴 ∩ 𝐶) ≠ ∅ → ((𝐴 × 𝐵) “ 𝐶) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ≠ wne 2932 ∩ cin 3852 ∅c0 4223 ifcif 4425 × cxp 5534 “ cima 5539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-xp 5542 df-rel 5543 df-cnv 5544 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 |
This theorem is referenced by: xpimasn 6028 ustuqtop1 23093 ustuqtop5 23097 brtrclfv2 40953 aacllem 46119 |
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