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Mirrors > Home > MPE Home > Th. List > xpsfrn | Structured version Visualization version GIF version |
Description: A short expression for the indexed cartesian product on two indices. (Contributed by Mario Carneiro, 15-Aug-2015.) |
Ref | Expression |
---|---|
xpsff1o.f | ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) |
Ref | Expression |
---|---|
xpsfrn | ⊢ ran 𝐹 = X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpsff1o.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) | |
2 | 1 | xpsff1o 17278 | . 2 ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) |
3 | f1ofo 6723 | . 2 ⊢ (𝐹:(𝐴 × 𝐵)–1-1-onto→X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → 𝐹:(𝐴 × 𝐵)–onto→X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)) | |
4 | forn 6691 | . 2 ⊢ (𝐹:(𝐴 × 𝐵)–onto→X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → ran 𝐹 = X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)) | |
5 | 2, 3, 4 | mp2b 10 | 1 ⊢ ran 𝐹 = X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ∅c0 4256 ifcif 4459 {cpr 4563 〈cop 4567 × cxp 5587 ran crn 5590 –onto→wfo 6431 –1-1-onto→wf1o 6432 ∈ cmpo 7277 1oc1o 8290 2oc2o 8291 Xcixp 8685 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-1o 8297 df-2o 8298 df-ixp 8686 df-en 8734 df-fin 8737 |
This theorem is referenced by: xpsrnbas 17282 |
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