Theorem smoeq 8280

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 22	. . . 4 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
2		dmeq 5845	. . . 4 ⊢ (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
3	1, 2	feq12d 6643	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴:dom 𝐴⟶On ↔ 𝐵:dom 𝐵⟶On))
4		ordeq 6317	. . . 4 ⊢ (dom 𝐴 = dom 𝐵 → (Ord dom 𝐴 ↔ Ord dom 𝐵))
5	2, 4	syl 17	. . 3 ⊢ (𝐴 = 𝐵 → (Ord dom 𝐴 ↔ Ord dom 𝐵))
6		fveq1 6826	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴‘𝑥) = (𝐵‘𝑥))
7		fveq1 6826	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴‘𝑦) = (𝐵‘𝑦))
8	6, 7	eleq12d 2833	. . . . . 6 ⊢ (𝐴 = 𝐵 → ((𝐴‘𝑥) ∈ (𝐴‘𝑦) ↔ (𝐵‘𝑥) ∈ (𝐵‘𝑦)))
9	8	imbi2d 341	. . . . 5 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)) ↔ (𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦))))
10	9	2ralbidv 3203	. . . 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 3297	. . . . 5 ⊢ (𝐴 = 𝐵 → (∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦)) ↔ ∀𝑦 ∈ dom 𝐵(𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦))))
12	11	ralbidv 3162	. . . 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 3297	. . . 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 306	. . 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 1444	. 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 8276	. 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 8276	. 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 315	1 ⊢ (𝐴 = 𝐵 → (Smo 𝐴 ↔ Smo 𝐵))

Syntax hints:   → wi 4   ↔ wb 207   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053  dom cdm 5618  Ord word 6309  Oncon0 6310  ⟶wf 6481  ‘cfv 6485  Smo wsmo 8275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-tr 5180  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-ord 6313  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fv 6493  df-smo 8276

This theorem is referenced by:  smores3  8283  smo0  8288  cofsmo  10182  cfsmolem  10183  alephsing  10189