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Theorem symgfvne 19367

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 19361	. . 3 ⊢ (𝐹 ∈ 𝐵 → 𝐹:𝐴–1-1-onto→𝐴)
4		f1of1 6822	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → 𝐹:𝐴–1-1→𝐴)
5		eqeq2 2748	. . . . . . . 8 ⊢ (𝑍 = (𝐹‘𝑋) → ((𝐹‘𝑌) = 𝑍 ↔ (𝐹‘𝑌) = (𝐹‘𝑋)))
6	5	eqcoms 2744	. . . . . . 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 1136	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1→𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝐹:𝐴–1-1→𝐴)
9		simp3 1138	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1→𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑌 ∈ 𝐴)
10		simp2 1137	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1→𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑋 ∈ 𝐴)
11		f1veqaeq 7254	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1→𝐴 ∧ (𝑌 ∈ 𝐴 ∧ 𝑋 ∈ 𝐴)) → ((𝐹‘𝑌) = (𝐹‘𝑋) → 𝑌 = 𝑋))
12	8, 9, 10, 11	syl12anc 836	. . . . . . 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 240	. . . . 5 ⊢ (((𝐹:𝐴–1-1→𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (𝐹‘𝑋) = 𝑍) → ((𝐹‘𝑌) = 𝑍 → 𝑌 = 𝑋))
15	14	necon3d 2954	. . . 4 ⊢ (((𝐹:𝐴–1-1→𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (𝐹‘𝑋) = 𝑍) → (𝑌 ≠ 𝑋 → (𝐹‘𝑌) ≠ 𝑍))
16	15	3exp1 1353	. . 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 1110	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2933 –1-1→wf1 6533 –1-1-onto→wf1o 6535 ‘cfv 6536 Basecbs 17233 SymGrpcsymg 19355

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8724 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-uz 12858 df-fz 13530 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-tset 17295 df-efmnd 18852 df-symg 19356

This theorem is referenced by: gsummatr01lem4 22601

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