Theorem smoel 8359

Description: If 𝑥 is less than 𝑦 then a strictly monotone function's value will be strictly less at 𝑥 than at 𝑦. (Contributed by Andrew Salmon, 22-Nov-2011.)

Assertion

Ref	Expression
smoel	⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐵‘𝐶) ∈ (𝐵‘𝐴))

Proof of Theorem smoel

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

Step	Hyp	Ref	Expression
1		smodm 8350	. . . . 5 ⊢ (Smo 𝐵 → Ord dom 𝐵)
2		ordtr1 6407	. . . . . . 7 ⊢ (Ord dom 𝐵 → ((𝐶 ∈ 𝐴 ∧ 𝐴 ∈ dom 𝐵) → 𝐶 ∈ dom 𝐵))
3	2	ancomsd 466	. . . . . 6 ⊢ (Ord dom 𝐵 → ((𝐴 ∈ dom 𝐵 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ dom 𝐵))
4	3	expdimp 453	. . . . 5 ⊢ ((Ord dom 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝐶 ∈ 𝐴 → 𝐶 ∈ dom 𝐵))
5	1, 4	sylan 580	. . . 4 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝐶 ∈ 𝐴 → 𝐶 ∈ dom 𝐵))
6		df-smo 8345	. . . . . 6 ⊢ (Smo 𝐵 ↔ (𝐵:dom 𝐵⟶On ∧ Ord dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐵∀𝑦 ∈ dom 𝐵(𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦))))
7		eleq1 2821	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝑦 ↔ 𝐶 ∈ 𝑦))
8		fveq2 6891	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐶 → (𝐵‘𝑥) = (𝐵‘𝐶))
9	8	eleq1d 2818	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐶 → ((𝐵‘𝑥) ∈ (𝐵‘𝑦) ↔ (𝐵‘𝐶) ∈ (𝐵‘𝑦)))
10	7, 9	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑥 = 𝐶 → ((𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦)) ↔ (𝐶 ∈ 𝑦 → (𝐵‘𝐶) ∈ (𝐵‘𝑦))))
11		eleq2 2822	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝐶 ∈ 𝑦 ↔ 𝐶 ∈ 𝐴))
12		fveq2 6891	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → (𝐵‘𝑦) = (𝐵‘𝐴))
13	12	eleq2d 2819	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → ((𝐵‘𝐶) ∈ (𝐵‘𝑦) ↔ (𝐵‘𝐶) ∈ (𝐵‘𝐴)))
14	11, 13	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → ((𝐶 ∈ 𝑦 → (𝐵‘𝐶) ∈ (𝐵‘𝑦)) ↔ (𝐶 ∈ 𝐴 → (𝐵‘𝐶) ∈ (𝐵‘𝐴))))
15	10, 14	rspc2v 3622	. . . . . . . . 9 ⊢ ((𝐶 ∈ dom 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (∀𝑥 ∈ dom 𝐵∀𝑦 ∈ dom 𝐵(𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦)) → (𝐶 ∈ 𝐴 → (𝐵‘𝐶) ∈ (𝐵‘𝐴))))
16	15	ancoms 459	. . . . . . . 8 ⊢ ((𝐴 ∈ dom 𝐵 ∧ 𝐶 ∈ dom 𝐵) → (∀𝑥 ∈ dom 𝐵∀𝑦 ∈ dom 𝐵(𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦)) → (𝐶 ∈ 𝐴 → (𝐵‘𝐶) ∈ (𝐵‘𝐴))))
17	16	com12 32	. . . . . . 7 ⊢ (∀𝑥 ∈ dom 𝐵∀𝑦 ∈ dom 𝐵(𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦)) → ((𝐴 ∈ dom 𝐵 ∧ 𝐶 ∈ dom 𝐵) → (𝐶 ∈ 𝐴 → (𝐵‘𝐶) ∈ (𝐵‘𝐴))))
18	17	3ad2ant3 1135	. . . . . 6 ⊢ ((𝐵:dom 𝐵⟶On ∧ Ord dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐵∀𝑦 ∈ dom 𝐵(𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦))) → ((𝐴 ∈ dom 𝐵 ∧ 𝐶 ∈ dom 𝐵) → (𝐶 ∈ 𝐴 → (𝐵‘𝐶) ∈ (𝐵‘𝐴))))
19	6, 18	sylbi 216	. . . . 5 ⊢ (Smo 𝐵 → ((𝐴 ∈ dom 𝐵 ∧ 𝐶 ∈ dom 𝐵) → (𝐶 ∈ 𝐴 → (𝐵‘𝐶) ∈ (𝐵‘𝐴))))
20	19	expdimp 453	. . . 4 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝐶 ∈ dom 𝐵 → (𝐶 ∈ 𝐴 → (𝐵‘𝐶) ∈ (𝐵‘𝐴))))
21	5, 20	syld 47	. . 3 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝐶 ∈ 𝐴 → (𝐶 ∈ 𝐴 → (𝐵‘𝐶) ∈ (𝐵‘𝐴))))
22	21	pm2.43d 53	. 2 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝐶 ∈ 𝐴 → (𝐵‘𝐶) ∈ (𝐵‘𝐴)))
23	22	3impia 1117	1 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐵‘𝐶) ∈ (𝐵‘𝐴))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061  dom cdm 5676  Ord word 6363  Oncon0 6364  ⟶wf 6539  ‘cfv 6543  Smo wsmo 8344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-tr 5266  df-ord 6367  df-iota 6495  df-fv 6551  df-smo 8345

This theorem is referenced by:  smoiun  8360  smoel2  8362