Theorem smoeq 7981

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 5766	. . . 4 ⊢ (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
3	1, 2	feq12d 6496	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴:dom 𝐴⟶On ↔ 𝐵:dom 𝐵⟶On))
4		ordeq 6192	. . . 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 6663	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴‘𝑥) = (𝐵‘𝑥))
7		fveq1 6663	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴‘𝑦) = (𝐵‘𝑦))
8	6, 7	eleq12d 2907	. . . . . 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 3199	. . . 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 3415	. . . . 5 ⊢ (𝐴 = 𝐵 → (∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦)) ↔ ∀𝑦 ∈ dom 𝐵(𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦))))
12	11	ralbidv 3197	. . . 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 3415	. . . 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 307	. . 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 1432	. 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 7977	. 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 7977	. 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 316	1 ⊢ (𝐴 = 𝐵 → (Smo 𝐴 ↔ Smo 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∀wral 3138  dom cdm 5549  Ord word 6184  Oncon0 6185  ⟶wf 6345  ‘cfv 6349  Smo wsmo 7976

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-tr 5165  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-ord 6188  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357  df-smo 7977

This theorem is referenced by:  smores3  7984  smo0  7989  cofsmo  9685  cfsmolem  9686  alephsing  9692