Theorem smoeq 8265

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 5838	. . . 4 ⊢ (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
3	1, 2	feq12d 6634	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴:dom 𝐴⟶On ↔ 𝐵:dom 𝐵⟶On))
4		ordeq 6308	. . . 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 6816	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴‘𝑥) = (𝐵‘𝑥))
7		fveq1 6816	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴‘𝑦) = (𝐵‘𝑦))
8	6, 7	eleq12d 2825	. . . . . 6 ⊢ (𝐴 = 𝐵 → ((𝐴‘𝑥) ∈ (𝐴‘𝑦) ↔ (𝐵‘𝑥) ∈ (𝐵‘𝑦)))
9	8	imbi2d 340	. . . . 5 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)) ↔ (𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦))))
10	9	2ralbidv 3196	. . . 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 3292	. . . . 5 ⊢ (𝐴 = 𝐵 → (∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦)) ↔ ∀𝑦 ∈ dom 𝐵(𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦))))
12	11	ralbidv 3155	. . . 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 3292	. . . 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 305	. . 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 1438	. 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 8261	. 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 8261	. 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 314	1 ⊢ (𝐴 = 𝐵 → (Smo 𝐴 ↔ Smo 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  dom cdm 5611  Ord word 6300  Oncon0 6301  ⟶wf 6472  ‘cfv 6476  Smo wsmo 8260

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-tr 5194  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-ord 6304  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fv 6484  df-smo 8261

This theorem is referenced by:  smores3  8268  smo0  8273  cofsmo  10155  cfsmolem  10156  alephsing  10162