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Theorem symgfvne 18903

Description: The function values of a permutation for different arguments are different. (Contributed by AV, 8-Jan-2019.)

**Hypotheses**
Ref	Expression
symgbas.1	⊢ 𝐺 = (SymGrp‘𝐴)
symgbas.2	⊢ 𝐵 = (Base‘𝐺)

**Assertion**
Ref	Expression
symgfvne	⊢ ((𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑍 → (𝑌 ≠ 𝑋 → (𝐹‘𝑌) ≠ 𝑍)))

**Proof of Theorem symgfvne**
Step	Hyp	Ref	Expression
1		symgbas.1	. . . 4 ⊢ 𝐺 = (SymGrp‘𝐴)
2		symgbas.2	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
3	1, 2	symgbasf1o 18897	. . 3 ⊢ (𝐹 ∈ 𝐵 → 𝐹:𝐴–1-1-onto→𝐴)
4		f1of1 6699	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → 𝐹:𝐴–1-1→𝐴)
5		eqeq2 2750	. . . . . . . 8 ⊢ (𝑍 = (𝐹‘𝑋) → ((𝐹‘𝑌) = 𝑍 ↔ (𝐹‘𝑌) = (𝐹‘𝑋)))
6	5	eqcoms 2746	. . . . . . 7 ⊢ ((𝐹‘𝑋) = 𝑍 → ((𝐹‘𝑌) = 𝑍 ↔ (𝐹‘𝑌) = (𝐹‘𝑋)))
7	6	adantl 481	. . . . . 6 ⊢ (((𝐹:𝐴–1-1→𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (𝐹‘𝑋) = 𝑍) → ((𝐹‘𝑌) = 𝑍 ↔ (𝐹‘𝑌) = (𝐹‘𝑋)))
8		simp1 1134	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1→𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝐹:𝐴–1-1→𝐴)
9		simp3 1136	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1→𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑌 ∈ 𝐴)
10		simp2 1135	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1→𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑋 ∈ 𝐴)
11		f1veqaeq 7111	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1→𝐴 ∧ (𝑌 ∈ 𝐴 ∧ 𝑋 ∈ 𝐴)) → ((𝐹‘𝑌) = (𝐹‘𝑋) → 𝑌 = 𝑋))
12	8, 9, 10, 11	syl12anc 833	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1→𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹‘𝑌) = (𝐹‘𝑋) → 𝑌 = 𝑋))
13	12	adantr 480	. . . . . 6 ⊢ (((𝐹:𝐴–1-1→𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (𝐹‘𝑋) = 𝑍) → ((𝐹‘𝑌) = (𝐹‘𝑋) → 𝑌 = 𝑋))
14	7, 13	sylbid 239	. . . . 5 ⊢ (((𝐹:𝐴–1-1→𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (𝐹‘𝑋) = 𝑍) → ((𝐹‘𝑌) = 𝑍 → 𝑌 = 𝑋))
15	14	necon3d 2963	. . . 4 ⊢ (((𝐹:𝐴–1-1→𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (𝐹‘𝑋) = 𝑍) → (𝑌 ≠ 𝑋 → (𝐹‘𝑌) ≠ 𝑍))
16	15	3exp1 1350	. . 3 ⊢ (𝐹:𝐴–1-1→𝐴 → (𝑋 ∈ 𝐴 → (𝑌 ∈ 𝐴 → ((𝐹‘𝑋) = 𝑍 → (𝑌 ≠ 𝑋 → (𝐹‘𝑌) ≠ 𝑍)))))
17	3, 4, 16	3syl 18	. 2 ⊢ (𝐹 ∈ 𝐵 → (𝑋 ∈ 𝐴 → (𝑌 ∈ 𝐴 → ((𝐹‘𝑋) = 𝑍 → (𝑌 ≠ 𝑋 → (𝐹‘𝑌) ≠ 𝑍)))))
18	17	3imp 1109	1 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑍 → (𝑌 ≠ 𝑋 → (𝐹‘𝑌) ≠ 𝑍)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 –1-1→wf1 6415 –1-1-onto→wf1o 6417 ‘cfv 6418 Basecbs 16840 SymGrpcsymg 18889

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-tset 16907 df-efmnd 18423 df-symg 18890

This theorem is referenced by: gsummatr01lem4 21715

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