Theorem smoeq 8396

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 23	. . . 4 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
2		dmeq 5897	. . . 4 ⊢ (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
3	1, 2	feq12d 6727	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴:dom 𝐴⟶On ↔ 𝐵:dom 𝐵⟶On))
4		ordeq 6395	. . . 4 ⊢ (dom 𝐴 = dom 𝐵 → (Ord dom 𝐴 ↔ Ord dom 𝐵))
5	2, 4	syl 18	. . 3 ⊢ (𝐴 = 𝐵 → (Ord dom 𝐴 ↔ Ord dom 𝐵))
6		fveq1 6915	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴‘𝑥) = (𝐵‘𝑥))
7		fveq1 6915	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴‘𝑦) = (𝐵‘𝑦))
8	6, 7	eleq12d 2858	. . . . . 6 ⊢ (𝐴 = 𝐵 → ((𝐴‘𝑥) ∈ (𝐴‘𝑦) ↔ (𝐵‘𝑥) ∈ (𝐵‘𝑦)))
9	8	imbi2d 343	. . . . 5 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)) ↔ (𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦))))
10	9	2ralbidv 3230	. . . 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 3323	. . . . 5 ⊢ (𝐴 = 𝐵 → (∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦)) ↔ ∀𝑦 ∈ dom 𝐵(𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦))))
12	11	ralbidv 3189	. . . 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 3323	. . . 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 308	. . 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 1465	. 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 8392	. 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 8392	. 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 317	1 ⊢ (𝐴 = 𝐵 → (Smo 𝐴 ↔ Smo 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  dom cdm 5658  Ord word 6387  Oncon0 6388  ⟶wf 6562  ‘cfv 6566  Smo wsmo 8391

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

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-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-tr 5219  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-ord 6391  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-smo 8392

This theorem is used by:  smores3  8399  smo0  8404  cofsmo  10386  cfsmolem  10387  alephsing  10393