Theorem smoel 8353

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 8344	. . . . 5 ⊢ (Smo 𝐵 → Ord dom 𝐵)
2		ordtr1 6409	. . . . . . 7 ⊢ (Ord dom 𝐵 → ((𝐶 ∈ 𝐴 ∧ 𝐴 ∈ dom 𝐵) → 𝐶 ∈ dom 𝐵))
3	2	ancomsd 471	. . . . . 6 ⊢ (Ord dom 𝐵 → ((𝐴 ∈ dom 𝐵 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ dom 𝐵))
4	3	expdimp 458	. . . . 5 ⊢ ((Ord dom 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝐶 ∈ 𝐴 → 𝐶 ∈ dom 𝐵))
5	1, 4	sylan 592	. . . 4 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝐶 ∈ 𝐴 → 𝐶 ∈ dom 𝐵))
6		df-smo 8339	. . . . . 6 ⊢ (Smo 𝐵 ↔ (𝐵:dom 𝐵⟶On ∧ Ord dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐵∀𝑦 ∈ dom 𝐵(𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦))))
7		eleq1 2853	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝑦 ↔ 𝐶 ∈ 𝑦))
8		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐶 → (𝐵‘𝑥) = (𝐵‘𝐶))
9	8	eleq1d 2850	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐶 → ((𝐵‘𝑥) ∈ (𝐵‘𝑦) ↔ (𝐵‘𝐶) ∈ (𝐵‘𝑦)))
10	7, 9	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑥 = 𝐶 → ((𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦)) ↔ (𝐶 ∈ 𝑦 → (𝐵‘𝐶) ∈ (𝐵‘𝑦))))
11		eleq2 2854	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝐶 ∈ 𝑦 ↔ 𝐶 ∈ 𝐴))
12		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → (𝐵‘𝑦) = (𝐵‘𝐴))
13	12	eleq2d 2851	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → ((𝐵‘𝐶) ∈ (𝐵‘𝑦) ↔ (𝐵‘𝐶) ∈ (𝐵‘𝐴)))
14	11, 13	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → ((𝐶 ∈ 𝑦 → (𝐵‘𝐶) ∈ (𝐵‘𝑦)) ↔ (𝐶 ∈ 𝐴 → (𝐵‘𝐶) ∈ (𝐵‘𝐴))))
15	10, 14	rspc2v 3594	. . . . . . . . 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 464	. . . . . . . 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 33	. . . . . . 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 1153	. . . . . 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 220	. . . . 5 ⊢ (Smo 𝐵 → ((𝐴 ∈ dom 𝐵 ∧ 𝐶 ∈ dom 𝐵) → (𝐶 ∈ 𝐴 → (𝐵‘𝐶) ∈ (𝐵‘𝐴))))
20	19	expdimp 458	. . . 4 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝐶 ∈ dom 𝐵 → (𝐶 ∈ 𝐴 → (𝐵‘𝐶) ∈ (𝐵‘𝐴))))
21	5, 20	syld 48	. . 3 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝐶 ∈ 𝐴 → (𝐶 ∈ 𝐴 → (𝐵‘𝐶) ∈ (𝐵‘𝐴))))
22	21	pm2.43d 54	. 2 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝐶 ∈ 𝐴 → (𝐵‘𝐶) ∈ (𝐵‘𝐴)))
23	22	3impia 1135	1 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐵‘𝐶) ∈ (𝐵‘𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081  dom cdm 5663  Ord word 6363  Oncon0 6364  ⟶wf 6536  ‘cfv 6540  Smo wsmo 8338

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

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 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-tr 5221  df-ord 6367  df-iota 6496  df-fv 6548  df-smo 8339

This theorem is used by:  smoiun  8354  smoel2  8356