Theorem elghomlem1OLD 37872

Description: Obsolete as of 15-Mar-2020. Lemma for elghomOLD 37874. (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 7925	. . 3 ⊢ (𝐺 ∈ GrpOp → ran 𝐺 ∈ V)
2		rnexg 7925	. . 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 7959	. . 3 ⊢ ((ran 𝐺 ∈ V ∧ ran 𝐻 ∈ V) → 𝑆 ∈ V)
5	1, 2, 4	syl2an 596	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → 𝑆 ∈ V)
6		rneq 5950	. . . . . 6 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
7	6	feq2d 6723	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑓:ran 𝑔⟶ran ℎ ↔ 𝑓:ran 𝐺⟶ran ℎ))
8		oveq 7437	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
9	8	fveq2d 6911	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑓‘(𝑥𝑔𝑦)) = (𝑓‘(𝑥𝐺𝑦)))
10	9	eqeq2d 2746	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (((𝑓‘𝑥)ℎ(𝑓‘𝑦)) = (𝑓‘(𝑥𝑔𝑦)) ↔ ((𝑓‘𝑥)ℎ(𝑓‘𝑦)) = (𝑓‘(𝑥𝐺𝑦))))
11	6, 10	raleqbidv 3344	. . . . . 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 3344	. . . . 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 2806	. . 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 5950	. . . . . . 7 ⊢ (ℎ = 𝐻 → ran ℎ = ran 𝐻)
16	15	feq3d 6724	. . . . . 6 ⊢ (ℎ = 𝐻 → (𝑓:ran 𝐺⟶ran ℎ ↔ 𝑓:ran 𝐺⟶ran 𝐻))
17		oveq 7437	. . . . . . . 8 ⊢ (ℎ = 𝐻 → ((𝑓‘𝑥)ℎ(𝑓‘𝑦)) = ((𝑓‘𝑥)𝐻(𝑓‘𝑦)))
18	17	eqeq1d 2737	. . . . . . 7 ⊢ (ℎ = 𝐻 → (((𝑓‘𝑥)ℎ(𝑓‘𝑦)) = (𝑓‘(𝑥𝐺𝑦)) ↔ ((𝑓‘𝑥)𝐻(𝑓‘𝑦)) = (𝑓‘(𝑥𝐺𝑦))))
19	18	2ralbidv 3219	. . . . . 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 2806	. . . 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 2793	. . 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 37871	. . 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 7592	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝑆 ∈ V) → (𝐺 GrpOpHom 𝐻) = 𝑆)
25	5, 24	mpd3an3 1461	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐺 GrpOpHom 𝐻) = 𝑆)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  {cab 2712  ∀wral 3059  Vcvv 3478  ran crn 5690  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  GrpOpcgr 30518   GrpOpHom cghomOLD 37870

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-ghomOLD 37871

This theorem is referenced by:  elghomlem2OLD  37873