Theorem elghomlem1OLD 38636

Description: Obsolete as of 15-Mar-2020. Lemma for elghomOLD 38638. (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 7900	. . 3 ⊢ (𝐺 ∈ GrpOp → ran 𝐺 ∈ V)
2		rnexg 7900	. . 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 7936	. . 3 ⊢ ((ran 𝐺 ∈ V ∧ ran 𝐻 ∈ V) → 𝑆 ∈ V)
5	1, 2, 4	syl2an 608	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → 𝑆 ∈ V)
6		rneq 5920	. . . . . 6 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
7	6	feq2d 6687	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑓:ran 𝑔⟶ran ℎ ↔ 𝑓:ran 𝐺⟶ran ℎ))
8		oveq 7420	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
9	8	fveq2d 6883	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑓‘(𝑥𝑔𝑦)) = (𝑓‘(𝑥𝐺𝑦)))
10	9	eqeq2d 2771	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (((𝑓‘𝑥)ℎ(𝑓‘𝑦)) = (𝑓‘(𝑥𝑔𝑦)) ↔ ((𝑓‘𝑥)ℎ(𝑓‘𝑦)) = (𝑓‘(𝑥𝐺𝑦))))
11	6, 10	raleqbidv 3334	. . . . . 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 3334	. . . . 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 644	. . . 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 2826	. . 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 5920	. . . . . . 7 ⊢ (ℎ = 𝐻 → ran ℎ = ran 𝐻)
16	15	feq3d 6688	. . . . . 6 ⊢ (ℎ = 𝐻 → (𝑓:ran 𝐺⟶ran ℎ ↔ 𝑓:ran 𝐺⟶ran 𝐻))
17		oveq 7420	. . . . . . . 8 ⊢ (ℎ = 𝐻 → ((𝑓‘𝑥)ℎ(𝑓‘𝑦)) = ((𝑓‘𝑥)𝐻(𝑓‘𝑦)))
18	17	eqeq1d 2762	. . . . . . 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 644	. . . . 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 2826	. . . 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 2813	. . 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 38635	. . 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 7573	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝑆 ∈ V) → (𝐺 GrpOpHom 𝐻) = 𝑆)
25	5, 24	mpd3an3 1491	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐺 GrpOpHom 𝐻) = 𝑆)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2738  ∀wral 3076  Vcvv 3450  ran crn 5656  ⟶wf 6529  ‘cfv 6533  (class class class)co 7414  GrpOpcgr 30971   GrpOpHom cghomOLD 38634

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-ghomOLD 38635

This theorem is used by:  elghomlem2OLD  38637