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Theorem gafo 18405
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 18404 . 2 ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)⟶𝑌)
3 gagrp 18401 . . . . . 6 ( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
43adantr 484 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥𝑌) → 𝐺 ∈ Grp)
5 eqid 2821 . . . . . 6 (0g𝐺) = (0g𝐺)
61, 5grpidcl 18110 . . . . 5 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝑋)
74, 6syl 17 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥𝑌) → (0g𝐺) ∈ 𝑋)
8 simpr 488 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥𝑌) → 𝑥𝑌)
95gagrpid 18403 . . . . 5 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥𝑌) → ((0g𝐺) 𝑥) = 𝑥)
109eqcomd 2827 . . . 4 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥𝑌) → 𝑥 = ((0g𝐺) 𝑥))
11 rspceov 7177 . . . 4 (((0g𝐺) ∈ 𝑋𝑥𝑌𝑥 = ((0g𝐺) 𝑥)) → ∃𝑦𝑋𝑧𝑌 𝑥 = (𝑦 𝑧))
127, 8, 10, 11syl3anc 1368 . . 3 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑥𝑌) → ∃𝑦𝑋𝑧𝑌 𝑥 = (𝑦 𝑧))
1312ralrimiva 3170 . 2 ( ∈ (𝐺 GrpAct 𝑌) → ∀𝑥𝑌𝑦𝑋𝑧𝑌 𝑥 = (𝑦 𝑧))
14 foov 7297 . 2 ( :(𝑋 × 𝑌)–onto𝑌 ↔ ( :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥𝑌𝑦𝑋𝑧𝑌 𝑥 = (𝑦 𝑧)))
152, 13, 14sylanbrc 586 1 ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)–onto𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3126  wrex 3127   × cxp 5526  wf 6324  ontowfo 6326  cfv 6328  (class class class)co 7130  Basecbs 16462  0gc0g 16692  Grpcgrp 18082   GrpAct cga 18398
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fo 6334  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-map 8383  df-0g 16694  df-mgm 17831  df-sgrp 17880  df-mnd 17891  df-grp 18085  df-ga 18399
This theorem is referenced by: (None)
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