Theorem elghomlem1OLD 37933

Description: Obsolete as of 15-Mar-2020. Lemma for elghomOLD 37935. (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 7832	. . 3 ⊢ (𝐺 ∈ GrpOp → ran 𝐺 ∈ V)
2		rnexg 7832	. . 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 7868	. . 3 ⊢ ((ran 𝐺 ∈ V ∧ ran 𝐻 ∈ V) → 𝑆 ∈ V)
5	1, 2, 4	syl2an 596	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → 𝑆 ∈ V)
6		rneq 5875	. . . . . 6 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
7	6	feq2d 6635	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑓:ran 𝑔⟶ran ℎ ↔ 𝑓:ran 𝐺⟶ran ℎ))
8		oveq 7352	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
9	8	fveq2d 6826	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑓‘(𝑥𝑔𝑦)) = (𝑓‘(𝑥𝐺𝑦)))
10	9	eqeq2d 2742	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (((𝑓‘𝑥)ℎ(𝑓‘𝑦)) = (𝑓‘(𝑥𝑔𝑦)) ↔ ((𝑓‘𝑥)ℎ(𝑓‘𝑦)) = (𝑓‘(𝑥𝐺𝑦))))
11	6, 10	raleqbidv 3312	. . . . . 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 3312	. . . . 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 632	. . . 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 2797	. . 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 5875	. . . . . . 7 ⊢ (ℎ = 𝐻 → ran ℎ = ran 𝐻)
16	15	feq3d 6636	. . . . . 6 ⊢ (ℎ = 𝐻 → (𝑓:ran 𝐺⟶ran ℎ ↔ 𝑓:ran 𝐺⟶ran 𝐻))
17		oveq 7352	. . . . . . . 8 ⊢ (ℎ = 𝐻 → ((𝑓‘𝑥)ℎ(𝑓‘𝑦)) = ((𝑓‘𝑥)𝐻(𝑓‘𝑦)))
18	17	eqeq1d 2733	. . . . . . 7 ⊢ (ℎ = 𝐻 → (((𝑓‘𝑥)ℎ(𝑓‘𝑦)) = (𝑓‘(𝑥𝐺𝑦)) ↔ ((𝑓‘𝑥)𝐻(𝑓‘𝑦)) = (𝑓‘(𝑥𝐺𝑦))))
19	18	2ralbidv 3196	. . . . . 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 632	. . . . 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 2797	. . . 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 2784	. . 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 37932	. . 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 7505	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝑆 ∈ V) → (𝐺 GrpOpHom 𝐻) = 𝑆)
25	5, 24	mpd3an3 1464	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐺 GrpOpHom 𝐻) = 𝑆)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {cab 2709  ∀wral 3047  Vcvv 3436  ran crn 5615  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  GrpOpcgr 30469   GrpOpHom cghomOLD 37931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-ghomOLD 37932

This theorem is referenced by:  elghomlem2OLD  37934