Theorem elghomlem2OLD 38221

Description: Obsolete as of 15-Mar-2020. Lemma for elghomOLD 38222. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
elghomlem1OLD.1	⊢ 𝑆 = {𝑓 ∣ (𝑓:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑓‘𝑥)𝐻(𝑓‘𝑦)) = (𝑓‘(𝑥𝐺𝑦)))}

Assertion

Ref	Expression
elghomlem2OLD	⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝐹:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))

Distinct variable groups:   𝑥,𝑓,𝑦,𝐹   𝑓,𝐺,𝑥,𝑦   𝑓,𝐻,𝑥,𝑦

Allowed substitution hints:   𝑆(𝑥,𝑦,𝑓)

Proof of Theorem elghomlem2OLD

Step	Hyp	Ref	Expression
1		elghomlem1OLD.1	. . . 4 ⊢ 𝑆 = {𝑓 ∣ (𝑓:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑓‘𝑥)𝐻(𝑓‘𝑦)) = (𝑓‘(𝑥𝐺𝑦)))}
2	1	elghomlem1OLD 38220	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐺 GrpOpHom 𝐻) = 𝑆)
3	2	eleq2d 2823	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ 𝐹 ∈ 𝑆))
4		elex 3451	. . . . 5 ⊢ (𝐹 ∈ 𝑆 → 𝐹 ∈ V)
5		feq1 6640	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓:ran 𝐺⟶ran 𝐻 ↔ 𝐹:ran 𝐺⟶ran 𝐻))
6		fveq1 6833	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
7		fveq1 6833	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦))
8	6, 7	oveq12d 7378	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥)𝐻(𝑓‘𝑦)) = ((𝐹‘𝑥)𝐻(𝐹‘𝑦)))
9		fveq1 6833	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥𝐺𝑦)) = (𝐹‘(𝑥𝐺𝑦)))
10	8, 9	eqeq12d 2753	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (((𝑓‘𝑥)𝐻(𝑓‘𝑦)) = (𝑓‘(𝑥𝐺𝑦)) ↔ ((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦))))
11	10	2ralbidv 3202	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑓‘𝑥)𝐻(𝑓‘𝑦)) = (𝑓‘(𝑥𝐺𝑦)) ↔ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦))))
12	5, 11	anbi12d 633	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑓:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑓‘𝑥)𝐻(𝑓‘𝑦)) = (𝑓‘(𝑥𝐺𝑦))) ↔ (𝐹:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))
13	12, 1	elab2g 3624	. . . . . 6 ⊢ (𝐹 ∈ V → (𝐹 ∈ 𝑆 ↔ (𝐹:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))
14	13	biimpd 229	. . . . 5 ⊢ (𝐹 ∈ V → (𝐹 ∈ 𝑆 → (𝐹:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))
15	4, 14	mpcom 38	. . . 4 ⊢ (𝐹 ∈ 𝑆 → (𝐹:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦))))
16		rnexg 7846	. . . . . . 7 ⊢ (𝐺 ∈ GrpOp → ran 𝐺 ∈ V)
17		fex 7174	. . . . . . . 8 ⊢ ((𝐹:ran 𝐺⟶ran 𝐻 ∧ ran 𝐺 ∈ V) → 𝐹 ∈ V)
18	17	expcom 413	. . . . . . 7 ⊢ (ran 𝐺 ∈ V → (𝐹:ran 𝐺⟶ran 𝐻 → 𝐹 ∈ V))
19	16, 18	syl 17	. . . . . 6 ⊢ (𝐺 ∈ GrpOp → (𝐹:ran 𝐺⟶ran 𝐻 → 𝐹 ∈ V))
20	19	adantrd 491	. . . . 5 ⊢ (𝐺 ∈ GrpOp → ((𝐹:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦))) → 𝐹 ∈ V))
21	13	biimprd 248	. . . . 5 ⊢ (𝐹 ∈ V → ((𝐹:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦))) → 𝐹 ∈ 𝑆))
22	20, 21	syli 39	. . . 4 ⊢ (𝐺 ∈ GrpOp → ((𝐹:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦))) → 𝐹 ∈ 𝑆))
23	15, 22	impbid2 226	. . 3 ⊢ (𝐺 ∈ GrpOp → (𝐹 ∈ 𝑆 ↔ (𝐹:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))
24	23	adantr 480	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐹 ∈ 𝑆 ↔ (𝐹:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))
25	3, 24	bitrd 279	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝐹:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  Vcvv 3430  ran crn 5625  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360  GrpOpcgr 30575   GrpOpHom cghomOLD 38218

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-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  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-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-ghomOLD 38219

This theorem is referenced by:  elghomOLD  38222