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Theorem iunfo 10483
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 7946 . . . 4 2nd :Vā€“ontoā†’V
2 fof 6760 . . . 4 (2nd :Vā€“ontoā†’V ā†’ 2nd :VāŸ¶V)
3 ffn 6672 . . . 4 (2nd :VāŸ¶V ā†’ 2nd Fn V)
41, 2, 3mp2b 10 . . 3 2nd Fn V
5 ssv 3972 . . 3 š‘‡ āŠ† V
6 fnssres 6628 . . 3 ((2nd Fn V āˆ§ š‘‡ āŠ† V) ā†’ (2nd ā†¾ š‘‡) Fn š‘‡)
74, 5, 6mp2an 691 . 2 (2nd ā†¾ š‘‡) Fn š‘‡
8 df-ima 5650 . . 3 (2nd ā€œ š‘‡) = ran (2nd ā†¾ š‘‡)
9 iunfo.1 . . . . . . . . . . 11 š‘‡ = āˆŖ š‘„ āˆˆ š“ ({š‘„} Ɨ šµ)
109eleq2i 2826 . . . . . . . . . 10 (š‘§ āˆˆ š‘‡ ā†” š‘§ āˆˆ āˆŖ š‘„ āˆˆ š“ ({š‘„} Ɨ šµ))
11 eliun 4962 . . . . . . . . . 10 (š‘§ āˆˆ āˆŖ š‘„ āˆˆ š“ ({š‘„} Ɨ šµ) ā†” āˆƒš‘„ āˆˆ š“ š‘§ āˆˆ ({š‘„} Ɨ šµ))
1210, 11bitri 275 . . . . . . . . 9 (š‘§ āˆˆ š‘‡ ā†” āˆƒš‘„ āˆˆ š“ š‘§ āˆˆ ({š‘„} Ɨ šµ))
13 xp2nd 7958 . . . . . . . . . . 11 (š‘§ āˆˆ ({š‘„} Ɨ šµ) ā†’ (2nd ā€˜š‘§) āˆˆ šµ)
14 eleq1 2822 . . . . . . . . . . 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 3164 . . . . . . . . 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 409 . . . . . . 7 ((š‘§ āˆˆ š‘‡ āˆ§ (2nd ā€˜š‘§) = š‘¦) ā†’ āˆƒš‘„ āˆˆ š“ š‘¦ āˆˆ šµ)
1918rexlimiva 3141 . . . . . 6 (āˆƒš‘§ āˆˆ š‘‡ (2nd ā€˜š‘§) = š‘¦ ā†’ āˆƒš‘„ āˆˆ š“ š‘¦ āˆˆ šµ)
20 nfiu1 4992 . . . . . . . . 9 ā„²š‘„āˆŖ š‘„ āˆˆ š“ ({š‘„} Ɨ šµ)
219, 20nfcxfr 2902 . . . . . . . 8 ā„²š‘„š‘‡
22 nfv 1918 . . . . . . . 8 ā„²š‘„(2nd ā€˜š‘§) = š‘¦
2321, 22nfrexw 3295 . . . . . . 7 ā„²š‘„āˆƒš‘§ āˆˆ š‘‡ (2nd ā€˜š‘§) = š‘¦
24 ssiun2 5011 . . . . . . . . . . . 12 (š‘„ āˆˆ š“ ā†’ ({š‘„} Ɨ šµ) āŠ† āˆŖ š‘„ āˆˆ š“ ({š‘„} Ɨ šµ))
2524adantr 482 . . . . . . . . . . 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 486 . . . . . . . . . . . 12 ((š‘„ āˆˆ š“ āˆ§ š‘¦ āˆˆ šµ) ā†’ š‘¦ āˆˆ šµ)
27 vsnid 4627 . . . . . . . . . . . . 13 š‘„ āˆˆ {š‘„}
28 opelxp 5673 . . . . . . . . . . . . 13 (āŸØš‘„, š‘¦āŸ© āˆˆ ({š‘„} Ɨ šµ) ā†” (š‘„ āˆˆ {š‘„} āˆ§ š‘¦ āˆˆ šµ))
2927, 28mpbiran 708 . . . . . . . . . . . 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 3949 . . . . . . . . . 10 ((š‘„ āˆˆ š“ āˆ§ š‘¦ āˆˆ šµ) ā†’ āŸØš‘„, š‘¦āŸ© āˆˆ āˆŖ š‘„ āˆˆ š“ ({š‘„} Ɨ šµ))
3231, 9eleqtrrdi 2845 . . . . . . . . 9 ((š‘„ āˆˆ š“ āˆ§ š‘¦ āˆˆ šµ) ā†’ āŸØš‘„, š‘¦āŸ© āˆˆ š‘‡)
33 vex 3451 . . . . . . . . . 10 š‘„ āˆˆ V
34 vex 3451 . . . . . . . . . 10 š‘¦ āˆˆ V
3533, 34op2nd 7934 . . . . . . . . 9 (2nd ā€˜āŸØš‘„, š‘¦āŸ©) = š‘¦
36 fveqeq2 6855 . . . . . . . . . 10 (š‘§ = āŸØš‘„, š‘¦āŸ© ā†’ ((2nd ā€˜š‘§) = š‘¦ ā†” (2nd ā€˜āŸØš‘„, š‘¦āŸ©) = š‘¦))
3736rspcev 3583 . . . . . . . . 9 ((āŸØš‘„, š‘¦āŸ© āˆˆ š‘‡ āˆ§ (2nd ā€˜āŸØš‘„, š‘¦āŸ©) = š‘¦) ā†’ āˆƒš‘§ āˆˆ š‘‡ (2nd ā€˜š‘§) = š‘¦)
3832, 35, 37sylancl 587 . . . . . . . 8 ((š‘„ āˆˆ š“ āˆ§ š‘¦ āˆˆ šµ) ā†’ āˆƒš‘§ āˆˆ š‘‡ (2nd ā€˜š‘§) = š‘¦)
3938ex 414 . . . . . . 7 (š‘„ āˆˆ š“ ā†’ (š‘¦ āˆˆ šµ ā†’ āˆƒš‘§ āˆˆ š‘‡ (2nd ā€˜š‘§) = š‘¦))
4023, 39rexlimi 3241 . . . . . 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 6918 . . . . . 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 691 . . . . 5 (š‘¦ āˆˆ (2nd ā€œ š‘‡) ā†” āˆƒš‘§ āˆˆ š‘‡ (2nd ā€˜š‘§) = š‘¦)
44 eliun 4962 . . . . 5 (š‘¦ āˆˆ āˆŖ š‘„ āˆˆ š“ šµ ā†” āˆƒš‘„ āˆˆ š“ š‘¦ āˆˆ šµ)
4541, 43, 443bitr4i 303 . . . 4 (š‘¦ āˆˆ (2nd ā€œ š‘‡) ā†” š‘¦ āˆˆ āˆŖ š‘„ āˆˆ š“ šµ)
4645eqriv 2730 . . 3 (2nd ā€œ š‘‡) = āˆŖ š‘„ āˆˆ š“ šµ
478, 46eqtr3i 2763 . 2 ran (2nd ā†¾ š‘‡) = āˆŖ š‘„ āˆˆ š“ šµ
48 df-fo 6506 . 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 710 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 397   = wceq 1542   āˆˆ wcel 2107  āˆƒwrex 3070  Vcvv 3447   āŠ† wss 3914  {csn 4590  āŸØcop 4596  āˆŖ ciun 4958   Ɨ cxp 5635  ran crn 5638   ā†¾ cres 5639   ā€œ cima 5640   Fn wfn 6495  āŸ¶wf 6496  ā€“ontoā†’wfo 6498  ā€˜cfv 6500  2nd c2nd 7924
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508  df-2nd 7926
This theorem is referenced by:  iundomg  10485  2ndresdjuf1o  31619
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