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Theorem xpsff1o2 16354
Description: The function appearing in xpsval 16355 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.)
Hypothesis
Ref Expression
xpsff1o.f 𝐹 = (𝑥𝐴, 𝑦𝐵({𝑥} +𝑐 {𝑦}))
Assertion
Ref Expression
xpsff1o2 𝐹:(𝐴 × 𝐵)–1-1-onto→ran 𝐹
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem xpsff1o2
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 xpsff1o.f . . 3 𝐹 = (𝑥𝐴, 𝑦𝐵({𝑥} +𝑐 {𝑦}))
21xpsff1o 16351 . 2 𝐹:(𝐴 × 𝐵)–1-1-ontoX𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵)
3 f1of1 6249 . 2 (𝐹:(𝐴 × 𝐵)–1-1-ontoX𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) → 𝐹:(𝐴 × 𝐵)–1-1X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵))
4 f1f1orn 6261 . 2 (𝐹:(𝐴 × 𝐵)–1-1X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) → 𝐹:(𝐴 × 𝐵)–1-1-onto→ran 𝐹)
52, 3, 4mp2b 10 1 𝐹:(𝐴 × 𝐵)–1-1-onto→ran 𝐹
Colors of variables: wff setvar class
Syntax hints:   = wceq 1596  c0 4023  ifcif 4194  {csn 4285   × cxp 5216  ccnv 5217  ran crn 5219  1-1wf1 5998  1-1-ontowf1o 6000  (class class class)co 6765  cmpt2 6767  2𝑜c2o 7674  Xcixp 8025   +𝑐 ccda 9102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-2o 7681  df-oadd 7684  df-er 7862  df-ixp 8026  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-cda 9103
This theorem is referenced by:  xpsbas  16357  xpsaddlem  16358  xpsadd  16359  xpsmul  16360  xpssca  16361  xpsvsca  16362  xpsless  16363  xpsle  16364  xpsmnd  17452  xpsgrp  17656  xpstps  21736  xpstopnlem2  21737  xpsdsfn  22304  xpsxmet  22307  xpsdsval  22308  xpsmet  22309  xpsxms  22461  xpsms  22462
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