Theorem elghomlem1OLD 38384

Description: Obsolete as of 15-Mar-2020. Lemma for elghomOLD 38386. (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
elghomlem1OLD	⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐺 GrpOpHom 𝐻) = 𝑆)

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

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

Proof of Theorem elghomlem1OLD

Dummy variables 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rnexg 7883	. . 3 ⊢ (𝐺 ∈ GrpOp → ran 𝐺 ∈ V)
2		rnexg 7883	. . 3 ⊢ (𝐻 ∈ GrpOp → ran 𝐻 ∈ V)
3		elghomlem1OLD.1	. . . 4 ⊢ 𝑆 = {𝑓 ∣ (𝑓:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑓‘𝑥)𝐻(𝑓‘𝑦)) = (𝑓‘(𝑥𝐺𝑦)))}
4	3	fabexg 7919	. . 3 ⊢ ((ran 𝐺 ∈ V ∧ ran 𝐻 ∈ V) → 𝑆 ∈ V)
5	1, 2, 4	syl2an 605	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → 𝑆 ∈ V)
6		rneq 5912	. . . . . 6 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
7	6	feq2d 6675	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑓:ran 𝑔⟶ran ℎ ↔ 𝑓:ran 𝐺⟶ran ℎ))
8		oveq 7402	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
9	8	fveq2d 6871	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑓‘(𝑥𝑔𝑦)) = (𝑓‘(𝑥𝐺𝑦)))
10	9	eqeq2d 2773	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (((𝑓‘𝑥)ℎ(𝑓‘𝑦)) = (𝑓‘(𝑥𝑔𝑦)) ↔ ((𝑓‘𝑥)ℎ(𝑓‘𝑦)) = (𝑓‘(𝑥𝐺𝑦))))
11	6, 10	raleqbidv 3336	. . . . . 6 ⊢ (𝑔 = 𝐺 → (∀𝑦 ∈ ran 𝑔((𝑓‘𝑥)ℎ(𝑓‘𝑦)) = (𝑓‘(𝑥𝑔𝑦)) ↔ ∀𝑦 ∈ ran 𝐺((𝑓‘𝑥)ℎ(𝑓‘𝑦)) = (𝑓‘(𝑥𝐺𝑦))))
12	6, 11	raleqbidv 3336	. . . . 5 ⊢ (𝑔 = 𝐺 → (∀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))))
13	7, 12	anbi12d 641	. . . 4 ⊢ (𝑔 = 𝐺 → ((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)))))
14	13	abbidv 2828	. . 3 ⊢ (𝑔 = 𝐺 → {𝑓 ∣ (𝑓:ran 𝑔⟶ran ℎ ∧ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔((𝑓‘𝑥)ℎ(𝑓‘𝑦)) = (𝑓‘(𝑥𝑔𝑦)))} = {𝑓 ∣ (𝑓:ran 𝐺⟶ran ℎ ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑓‘𝑥)ℎ(𝑓‘𝑦)) = (𝑓‘(𝑥𝐺𝑦)))})
15		rneq 5912	. . . . . . 7 ⊢ (ℎ = 𝐻 → ran ℎ = ran 𝐻)
16	15	feq3d 6676	. . . . . 6 ⊢ (ℎ = 𝐻 → (𝑓:ran 𝐺⟶ran ℎ ↔ 𝑓:ran 𝐺⟶ran 𝐻))
17		oveq 7402	. . . . . . . 8 ⊢ (ℎ = 𝐻 → ((𝑓‘𝑥)ℎ(𝑓‘𝑦)) = ((𝑓‘𝑥)𝐻(𝑓‘𝑦)))
18	17	eqeq1d 2764	. . . . . . 7 ⊢ (ℎ = 𝐻 → (((𝑓‘𝑥)ℎ(𝑓‘𝑦)) = (𝑓‘(𝑥𝐺𝑦)) ↔ ((𝑓‘𝑥)𝐻(𝑓‘𝑦)) = (𝑓‘(𝑥𝐺𝑦))))
19	18	2ralbidv 3226	. . . . . 6 ⊢ (ℎ = 𝐻 → (∀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))))
20	16, 19	anbi12d 641	. . . . 5 ⊢ (ℎ = 𝐻 → ((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)))))
21	20	abbidv 2828	. . . 4 ⊢ (ℎ = 𝐻 → {𝑓 ∣ (𝑓:ran 𝐺⟶ran ℎ ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑓‘𝑥)ℎ(𝑓‘𝑦)) = (𝑓‘(𝑥𝐺𝑦)))} = {𝑓 ∣ (𝑓:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑓‘𝑥)𝐻(𝑓‘𝑦)) = (𝑓‘(𝑥𝐺𝑦)))})
22	21, 3	eqtr4di 2815	. . 3 ⊢ (ℎ = 𝐻 → {𝑓 ∣ (𝑓:ran 𝐺⟶ran ℎ ∧ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑓‘𝑥)ℎ(𝑓‘𝑦)) = (𝑓‘(𝑥𝐺𝑦)))} = 𝑆)
23		df-ghomOLD 38383	. . 3 ⊢ GrpOpHom = (𝑔 ∈ GrpOp, ℎ ∈ GrpOp ↦ {𝑓 ∣ (𝑓:ran 𝑔⟶ran ℎ ∧ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔((𝑓‘𝑥)ℎ(𝑓‘𝑦)) = (𝑓‘(𝑥𝑔𝑦)))})
24	14, 22, 23	ovmpog 7555	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝑆 ∈ V) → (𝐺 GrpOpHom 𝐻) = 𝑆)
25	5, 24	mpd3an3 1483	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐺 GrpOpHom 𝐻) = 𝑆)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  {cab 2740  ∀wral 3076  Vcvv 3454  ran crn 5648  ⟶wf 6517  ‘cfv 6521  (class class class)co 7396  GrpOpcgr 30692   GrpOpHom cghomOLD 38382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-ghomOLD 38383

This theorem is referenced by:  elghomlem2OLD  38385