Theorem smoel 6319

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 6310	. . . . 5 ⊢ (Smo 𝐵 → Ord dom 𝐵)
2		ordtr1 4403	. . . . . . 7 ⊢ (Ord dom 𝐵 → ((𝐶 ∈ 𝐴 ∧ 𝐴 ∈ dom 𝐵) → 𝐶 ∈ dom 𝐵))
3	2	ancomsd 269	. . . . . 6 ⊢ (Ord dom 𝐵 → ((𝐴 ∈ dom 𝐵 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ dom 𝐵))
4	3	expdimp 259	. . . . 5 ⊢ ((Ord dom 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝐶 ∈ 𝐴 → 𝐶 ∈ dom 𝐵))
5	1, 4	sylan 283	. . . 4 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝐶 ∈ 𝐴 → 𝐶 ∈ dom 𝐵))
6		df-smo 6305	. . . . . 6 ⊢ (Smo 𝐵 ↔ (𝐵:dom 𝐵⟶On ∧ Ord dom 𝐵 ∧ ∀𝑥 ∈ dom 𝐵∀𝑦 ∈ dom 𝐵(𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦))))
7		eleq1 2252	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝑦 ↔ 𝐶 ∈ 𝑦))
8		fveq2 5530	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐶 → (𝐵‘𝑥) = (𝐵‘𝐶))
9	8	eleq1d 2258	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐶 → ((𝐵‘𝑥) ∈ (𝐵‘𝑦) ↔ (𝐵‘𝐶) ∈ (𝐵‘𝑦)))
10	7, 9	imbi12d 234	. . . . . . . . . 10 ⊢ (𝑥 = 𝐶 → ((𝑥 ∈ 𝑦 → (𝐵‘𝑥) ∈ (𝐵‘𝑦)) ↔ (𝐶 ∈ 𝑦 → (𝐵‘𝐶) ∈ (𝐵‘𝑦))))
11		eleq2 2253	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝐶 ∈ 𝑦 ↔ 𝐶 ∈ 𝐴))
12		fveq2 5530	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → (𝐵‘𝑦) = (𝐵‘𝐴))
13	12	eleq2d 2259	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → ((𝐵‘𝐶) ∈ (𝐵‘𝑦) ↔ (𝐵‘𝐶) ∈ (𝐵‘𝐴)))
14	11, 13	imbi12d 234	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → ((𝐶 ∈ 𝑦 → (𝐵‘𝐶) ∈ (𝐵‘𝑦)) ↔ (𝐶 ∈ 𝐴 → (𝐵‘𝐶) ∈ (𝐵‘𝐴))))
15	10, 14	rspc2v 2869	. . . . . . . . 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 268	. . . . . . . 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 30	. . . . . . 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 1022	. . . . . 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 121	. . . . 5 ⊢ (Smo 𝐵 → ((𝐴 ∈ dom 𝐵 ∧ 𝐶 ∈ dom 𝐵) → (𝐶 ∈ 𝐴 → (𝐵‘𝐶) ∈ (𝐵‘𝐴))))
20	19	expdimp 259	. . . 4 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝐶 ∈ dom 𝐵 → (𝐶 ∈ 𝐴 → (𝐵‘𝐶) ∈ (𝐵‘𝐴))))
21	5, 20	syld 45	. . 3 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝐶 ∈ 𝐴 → (𝐶 ∈ 𝐴 → (𝐵‘𝐶) ∈ (𝐵‘𝐴))))
22	21	pm2.43d 50	. 2 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝐶 ∈ 𝐴 → (𝐵‘𝐶) ∈ (𝐵‘𝐴)))
23	22	3impia 1202	1 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐵‘𝐶) ∈ (𝐵‘𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1364   ∈ wcel 2160  ∀wral 2468  Ord word 4377  Oncon0 4378  dom cdm 4641  ⟶wf 5227  ‘cfv 5231  Smo wsmo 6304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-tr 4117  df-iord 4381  df-iota 5193  df-fv 5239  df-smo 6305

This theorem is referenced by:  smoiun  6320  smoel2  6322