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Theorem gafo 17669
Description: A group action is onto its base set. (Contributed by Jeff Hankins, 10-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
Hypothesis
Ref Expression
gaf.1 𝑋 = (Base‘𝐺)
Assertion
Ref Expression
gafo ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)–onto𝑌)

Proof of Theorem gafo
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gaf.1 . . 3 𝑋 = (Base‘𝐺)
21gaf 17668 . 2 ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)⟶𝑌)
3 gagrp 17665 . . . . . 6 ( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
43adantr 481 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥𝑌) → 𝐺 ∈ Grp)
5 eqid 2621 . . . . . 6 (0g𝐺) = (0g𝐺)
61, 5grpidcl 17390 . . . . 5 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝑋)
74, 6syl 17 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥𝑌) → (0g𝐺) ∈ 𝑋)
8 simpr 477 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥𝑌) → 𝑥𝑌)
95gagrpid 17667 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥𝑌) → ((0g𝐺) 𝑥) = 𝑥)
109eqcomd 2627 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥𝑌) → 𝑥 = ((0g𝐺) 𝑥))
11 rspceov 6657 . . . 4 (((0g𝐺) ∈ 𝑋𝑥𝑌𝑥 = ((0g𝐺) 𝑥)) → ∃𝑦𝑋𝑧𝑌 𝑥 = (𝑦 𝑧))
127, 8, 10, 11syl3anc 1323 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥𝑌) → ∃𝑦𝑋𝑧𝑌 𝑥 = (𝑦 𝑧))
1312ralrimiva 2962 . 2 ( ∈ (𝐺 GrpAct 𝑌) → ∀𝑥𝑌𝑦𝑋𝑧𝑌 𝑥 = (𝑦 𝑧))
14 foov 6773 . 2 ( :(𝑋 × 𝑌)–onto𝑌 ↔ ( :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥𝑌𝑦𝑋𝑧𝑌 𝑥 = (𝑦 𝑧)))
152, 13, 14sylanbrc 697 1 ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)–onto𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2908  wrex 2909   × cxp 5082  wf 5853  ontowfo 5855  cfv 5857  (class class class)co 6615  Basecbs 15800  0gc0g 16040  Grpcgrp 17362   GrpAct cga 17662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fo 5863  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-map 7819  df-0g 16042  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-grp 17365  df-ga 17663
This theorem is referenced by: (None)
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