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Theorem symgfvne 19322

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 19316	. . 3 ⊢ (𝐹 ∈ 𝐵 → 𝐹:𝐴–1-1-onto→𝐴)
4		f1of1 6781	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → 𝐹:𝐴–1-1→𝐴)
5		eqeq2 2749	. . . . . . . 8 ⊢ (𝑍 = (𝐹‘𝑋) → ((𝐹‘𝑌) = 𝑍 ↔ (𝐹‘𝑌) = (𝐹‘𝑋)))
6	5	eqcoms 2745	. . . . . . 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 1137	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1→𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝐹:𝐴–1-1→𝐴)
9		simp3 1139	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1→𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑌 ∈ 𝐴)
10		simp2 1138	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1→𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑋 ∈ 𝐴)
11		f1veqaeq 7212	. . . . . . . 8 ⊢ ((𝐹:𝐴–1-1→𝐴 ∧ (𝑌 ∈ 𝐴 ∧ 𝑋 ∈ 𝐴)) → ((𝐹‘𝑌) = (𝐹‘𝑋) → 𝑌 = 𝑋))
12	8, 9, 10, 11	syl12anc 837	. . . . . . 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 1354	. . 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 1111	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 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 –1-1→wf1 6497 –1-1-onto→wf1o 6499 ‘cfv 6500 Basecbs 17148 SymGrpcsymg 19310

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 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-uz 12764 df-fz 13436 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-tset 17208 df-efmnd 18806 df-symg 19311

This theorem is referenced by: gsummatr01lem4 22614

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