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Mirrors > Home > MPE Home > Th. List > xpsff1o2 | Structured version Visualization version GIF version |
Description: The function appearing in xpsval 16055 is a bijection from the cartesian product to the indexed cartesian product indexed on the pair 2𝑜 = {∅, 1𝑜}. (Contributed by Mario Carneiro, 24-Jan-2015.) |
Ref | Expression |
---|---|
xpsff1o.f | ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ◡({𝑥} +𝑐 {𝑦})) |
Ref | Expression |
---|---|
xpsff1o2 | ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→ran 𝐹 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpsff1o.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ◡({𝑥} +𝑐 {𝑦})) | |
2 | 1 | xpsff1o 16051 | . 2 ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) |
3 | f1of1 6049 | . 2 ⊢ (𝐹:(𝐴 × 𝐵)–1-1-onto→X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) → 𝐹:(𝐴 × 𝐵)–1-1→X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵)) | |
4 | f1f1orn 6061 | . 2 ⊢ (𝐹:(𝐴 × 𝐵)–1-1→X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) → 𝐹:(𝐴 × 𝐵)–1-1-onto→ran 𝐹) | |
5 | 2, 3, 4 | mp2b 10 | 1 ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→ran 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∅c0 3874 ifcif 4036 {csn 4125 × cxp 5036 ◡ccnv 5037 ran crn 5039 –1-1→wf1 5801 –1-1-onto→wf1o 5803 (class class class)co 6549 ↦ cmpt2 6551 2𝑜c2o 7441 Xcixp 7794 +𝑐 ccda 8872 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-cda 8873 |
This theorem is referenced by: xpsbas 16057 xpsaddlem 16058 xpsadd 16059 xpsmul 16060 xpssca 16061 xpsvsca 16062 xpsless 16063 xpsle 16064 xpsmnd 17153 xpsgrp 17357 xpstps 21423 xpstopnlem2 21424 xpsdsfn 21992 xpsxmet 21995 xpsdsval 21996 xpsmet 21997 xpsxms 22149 xpsms 22150 |
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