Theorem smoeq 6290

Description: Equality theorem for strictly monotone functions. (Contributed by Andrew Salmon, 16-Nov-2011.)

Assertion

Ref	Expression
smoeq	⊢ (𝐴 = 𝐵 → (Smo 𝐴 ↔ Smo 𝐵))

Proof of Theorem smoeq

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 19	. . . 4 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
2		dmeq 4827	. . . 4 ⊢ (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
3	1, 2	feq12d 5355	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴:dom 𝐴⟶On ↔ 𝐵:dom 𝐵⟶On))
4		ordeq 4372	. . . 4 ⊢ (dom 𝐴 = dom 𝐵 → (Ord dom 𝐴 ↔ Ord dom 𝐵))
5	2, 4	syl 14	. . 3 ⊢ (𝐴 = 𝐵 → (Ord dom 𝐴 ↔ Ord dom 𝐵))
6		fveq1 5514	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴‘𝑥) = (𝐵‘𝑥))
7		fveq1 5514	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴‘𝑦) = (𝐵‘𝑦))
8	6, 7	eleq12d 2248	. . . . . 6 ⊢ (𝐴 = 𝐵 → ((𝐴‘𝑥) ∈ (𝐴‘𝑦) ↔ (𝐵‘𝑥) ∈ (𝐵‘𝑦)))
9	8	imbi2d 230	. . . . 5 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)) ↔ (𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦))))
10	9	2ralbidv 2501	. . . 4 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)) ↔ ∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦))))
11	2	raleqdv 2678	. . . . 5 ⊢ (𝐴 = 𝐵 → (∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦)) ↔ ∀𝑦 ∈ dom 𝐵(𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦))))
12	11	ralbidv 2477	. . . 4 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦)) ↔ ∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐵(𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦))))
13	2	raleqdv 2678	. . . 4 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐵(𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦)) ↔ ∀𝑥 ∈ dom 𝐵∀𝑦 ∈ dom 𝐵(𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦))))
14	10, 12, 13	3bitrd 214	. . 3 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)) ↔ ∀𝑥 ∈ dom 𝐵∀𝑦 ∈ dom 𝐵(𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦))))
15	3, 5, 14	3anbi123d 1312	. 2 ⊢ (𝐴 = 𝐵 → ((𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦))) ↔ (𝐵:dom 𝐵⟶On ∧ Ord dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐵∀𝑦 ∈ dom 𝐵(𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦)))))
16		df-smo 6286	. 2 ⊢ (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦))))
17		df-smo 6286	. 2 ⊢ (Smo 𝐵 ↔ (𝐵:dom 𝐵⟶On ∧ Ord dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐵∀𝑦 ∈ dom 𝐵(𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦))))
18	15, 16, 17	3bitr4g 223	1 ⊢ (𝐴 = 𝐵 → (Smo 𝐴 ↔ Smo 𝐵))

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  ∀wral 2455  Ord word 4362  Oncon0 4363  dom cdm 4626  ⟶wf 5212  ‘cfv 5216  Smo wsmo 6285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-tr 4102  df-iord 4366  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fv 5224  df-smo 6286

This theorem is referenced by:  smores3  6293  smo0  6298