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Theorem iunfo 10533
Description: Existence of an onto function from a disjoint union to a union. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 18-Jan-2014.)
Hypothesis
Ref Expression
iunfo.1 š‘‡ = āˆŖ š‘„ āˆˆ š“ ({š‘„} Ɨ šµ)
Assertion
Ref Expression
iunfo (2nd ā†¾ š‘‡):š‘‡ā€“ontoā†’āˆŖ š‘„ āˆˆ š“ šµ
Distinct variable group:   š‘„,š“
Allowed substitution hints:   šµ(š‘„)   š‘‡(š‘„)

Proof of Theorem iunfo
Dummy variables š‘¦ š‘§ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fo2nd 7995 . . . 4 2nd :Vā€“ontoā†’V
2 fof 6805 . . . 4 (2nd :Vā€“ontoā†’V ā†’ 2nd :VāŸ¶V)
3 ffn 6717 . . . 4 (2nd :VāŸ¶V ā†’ 2nd Fn V)
41, 2, 3mp2b 10 . . 3 2nd Fn V
5 ssv 4006 . . 3 š‘‡ āŠ† V
6 fnssres 6673 . . 3 ((2nd Fn V āˆ§ š‘‡ āŠ† V) ā†’ (2nd ā†¾ š‘‡) Fn š‘‡)
74, 5, 6mp2an 690 . 2 (2nd ā†¾ š‘‡) Fn š‘‡
8 df-ima 5689 . . 3 (2nd ā€œ š‘‡) = ran (2nd ā†¾ š‘‡)
9 iunfo.1 . . . . . . . . . . 11 š‘‡ = āˆŖ š‘„ āˆˆ š“ ({š‘„} Ɨ šµ)
109eleq2i 2825 . . . . . . . . . 10 (š‘§ āˆˆ š‘‡ ā†” š‘§ āˆˆ āˆŖ š‘„ āˆˆ š“ ({š‘„} Ɨ šµ))
11 eliun 5001 . . . . . . . . . 10 (š‘§ āˆˆ āˆŖ š‘„ āˆˆ š“ ({š‘„} Ɨ šµ) ā†” āˆƒš‘„ āˆˆ š“ š‘§ āˆˆ ({š‘„} Ɨ šµ))
1210, 11bitri 274 . . . . . . . . 9 (š‘§ āˆˆ š‘‡ ā†” āˆƒš‘„ āˆˆ š“ š‘§ āˆˆ ({š‘„} Ɨ šµ))
13 xp2nd 8007 . . . . . . . . . . 11 (š‘§ āˆˆ ({š‘„} Ɨ šµ) ā†’ (2nd ā€˜š‘§) āˆˆ šµ)
14 eleq1 2821 . . . . . . . . . . 11 ((2nd ā€˜š‘§) = š‘¦ ā†’ ((2nd ā€˜š‘§) āˆˆ šµ ā†” š‘¦ āˆˆ šµ))
1513, 14imbitrid 243 . . . . . . . . . 10 ((2nd ā€˜š‘§) = š‘¦ ā†’ (š‘§ āˆˆ ({š‘„} Ɨ šµ) ā†’ š‘¦ āˆˆ šµ))
1615reximdv 3170 . . . . . . . . 9 ((2nd ā€˜š‘§) = š‘¦ ā†’ (āˆƒš‘„ āˆˆ š“ š‘§ āˆˆ ({š‘„} Ɨ šµ) ā†’ āˆƒš‘„ āˆˆ š“ š‘¦ āˆˆ šµ))
1712, 16biimtrid 241 . . . . . . . 8 ((2nd ā€˜š‘§) = š‘¦ ā†’ (š‘§ āˆˆ š‘‡ ā†’ āˆƒš‘„ āˆˆ š“ š‘¦ āˆˆ šµ))
1817impcom 408 . . . . . . 7 ((š‘§ āˆˆ š‘‡ āˆ§ (2nd ā€˜š‘§) = š‘¦) ā†’ āˆƒš‘„ āˆˆ š“ š‘¦ āˆˆ šµ)
1918rexlimiva 3147 . . . . . 6 (āˆƒš‘§ āˆˆ š‘‡ (2nd ā€˜š‘§) = š‘¦ ā†’ āˆƒš‘„ āˆˆ š“ š‘¦ āˆˆ šµ)
20 nfiu1 5031 . . . . . . . . 9 ā„²š‘„āˆŖ š‘„ āˆˆ š“ ({š‘„} Ɨ šµ)
219, 20nfcxfr 2901 . . . . . . . 8 ā„²š‘„š‘‡
22 nfv 1917 . . . . . . . 8 ā„²š‘„(2nd ā€˜š‘§) = š‘¦
2321, 22nfrexw 3310 . . . . . . 7 ā„²š‘„āˆƒš‘§ āˆˆ š‘‡ (2nd ā€˜š‘§) = š‘¦
24 ssiun2 5050 . . . . . . . . . . . 12 (š‘„ āˆˆ š“ ā†’ ({š‘„} Ɨ šµ) āŠ† āˆŖ š‘„ āˆˆ š“ ({š‘„} Ɨ šµ))
2524adantr 481 . . . . . . . . . . 11 ((š‘„ āˆˆ š“ āˆ§ š‘¦ āˆˆ šµ) ā†’ ({š‘„} Ɨ šµ) āŠ† āˆŖ š‘„ āˆˆ š“ ({š‘„} Ɨ šµ))
26 simpr 485 . . . . . . . . . . . 12 ((š‘„ āˆˆ š“ āˆ§ š‘¦ āˆˆ šµ) ā†’ š‘¦ āˆˆ šµ)
27 vsnid 4665 . . . . . . . . . . . . 13 š‘„ āˆˆ {š‘„}
28 opelxp 5712 . . . . . . . . . . . . 13 (āŸØš‘„, š‘¦āŸ© āˆˆ ({š‘„} Ɨ šµ) ā†” (š‘„ āˆˆ {š‘„} āˆ§ š‘¦ āˆˆ šµ))
2927, 28mpbiran 707 . . . . . . . . . . . 12 (āŸØš‘„, š‘¦āŸ© āˆˆ ({š‘„} Ɨ šµ) ā†” š‘¦ āˆˆ šµ)
3026, 29sylibr 233 . . . . . . . . . . 11 ((š‘„ āˆˆ š“ āˆ§ š‘¦ āˆˆ šµ) ā†’ āŸØš‘„, š‘¦āŸ© āˆˆ ({š‘„} Ɨ šµ))
3125, 30sseldd 3983 . . . . . . . . . 10 ((š‘„ āˆˆ š“ āˆ§ š‘¦ āˆˆ šµ) ā†’ āŸØš‘„, š‘¦āŸ© āˆˆ āˆŖ š‘„ āˆˆ š“ ({š‘„} Ɨ šµ))
3231, 9eleqtrrdi 2844 . . . . . . . . 9 ((š‘„ āˆˆ š“ āˆ§ š‘¦ āˆˆ šµ) ā†’ āŸØš‘„, š‘¦āŸ© āˆˆ š‘‡)
33 vex 3478 . . . . . . . . . 10 š‘„ āˆˆ V
34 vex 3478 . . . . . . . . . 10 š‘¦ āˆˆ V
3533, 34op2nd 7983 . . . . . . . . 9 (2nd ā€˜āŸØš‘„, š‘¦āŸ©) = š‘¦
36 fveqeq2 6900 . . . . . . . . . 10 (š‘§ = āŸØš‘„, š‘¦āŸ© ā†’ ((2nd ā€˜š‘§) = š‘¦ ā†” (2nd ā€˜āŸØš‘„, š‘¦āŸ©) = š‘¦))
3736rspcev 3612 . . . . . . . . 9 ((āŸØš‘„, š‘¦āŸ© āˆˆ š‘‡ āˆ§ (2nd ā€˜āŸØš‘„, š‘¦āŸ©) = š‘¦) ā†’ āˆƒš‘§ āˆˆ š‘‡ (2nd ā€˜š‘§) = š‘¦)
3832, 35, 37sylancl 586 . . . . . . . 8 ((š‘„ āˆˆ š“ āˆ§ š‘¦ āˆˆ šµ) ā†’ āˆƒš‘§ āˆˆ š‘‡ (2nd ā€˜š‘§) = š‘¦)
3938ex 413 . . . . . . 7 (š‘„ āˆˆ š“ ā†’ (š‘¦ āˆˆ šµ ā†’ āˆƒš‘§ āˆˆ š‘‡ (2nd ā€˜š‘§) = š‘¦))
4023, 39rexlimi 3256 . . . . . 6 (āˆƒš‘„ āˆˆ š“ š‘¦ āˆˆ šµ ā†’ āˆƒš‘§ āˆˆ š‘‡ (2nd ā€˜š‘§) = š‘¦)
4119, 40impbii 208 . . . . 5 (āˆƒš‘§ āˆˆ š‘‡ (2nd ā€˜š‘§) = š‘¦ ā†” āˆƒš‘„ āˆˆ š“ š‘¦ āˆˆ šµ)
42 fvelimab 6964 . . . . . 6 ((2nd Fn V āˆ§ š‘‡ āŠ† V) ā†’ (š‘¦ āˆˆ (2nd ā€œ š‘‡) ā†” āˆƒš‘§ āˆˆ š‘‡ (2nd ā€˜š‘§) = š‘¦))
434, 5, 42mp2an 690 . . . . 5 (š‘¦ āˆˆ (2nd ā€œ š‘‡) ā†” āˆƒš‘§ āˆˆ š‘‡ (2nd ā€˜š‘§) = š‘¦)
44 eliun 5001 . . . . 5 (š‘¦ āˆˆ āˆŖ š‘„ āˆˆ š“ šµ ā†” āˆƒš‘„ āˆˆ š“ š‘¦ āˆˆ šµ)
4541, 43, 443bitr4i 302 . . . 4 (š‘¦ āˆˆ (2nd ā€œ š‘‡) ā†” š‘¦ āˆˆ āˆŖ š‘„ āˆˆ š“ šµ)
4645eqriv 2729 . . 3 (2nd ā€œ š‘‡) = āˆŖ š‘„ āˆˆ š“ šµ
478, 46eqtr3i 2762 . 2 ran (2nd ā†¾ š‘‡) = āˆŖ š‘„ āˆˆ š“ šµ
48 df-fo 6549 . 2 ((2nd ā†¾ š‘‡):š‘‡ā€“ontoā†’āˆŖ š‘„ āˆˆ š“ šµ ā†” ((2nd ā†¾ š‘‡) Fn š‘‡ āˆ§ ran (2nd ā†¾ š‘‡) = āˆŖ š‘„ āˆˆ š“ šµ))
497, 47, 48mpbir2an 709 1 (2nd ā†¾ š‘‡):š‘‡ā€“ontoā†’āˆŖ š‘„ āˆˆ š“ šµ
Colors of variables: wff setvar class
Syntax hints:   ā†” wb 205   āˆ§ wa 396   = wceq 1541   āˆˆ wcel 2106  āˆƒwrex 3070  Vcvv 3474   āŠ† wss 3948  {csn 4628  āŸØcop 4634  āˆŖ ciun 4997   Ɨ cxp 5674  ran crn 5677   ā†¾ cres 5678   ā€œ cima 5679   Fn wfn 6538  āŸ¶wf 6539  ā€“ontoā†’wfo 6541  ā€˜cfv 6543  2nd c2nd 7973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-2nd 7975
This theorem is referenced by:  iundomg  10535  2ndresdjuf1o  31870
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